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Derivatives: Logarithmic Functions

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Wize University Calculus 1 Textbook > Derivatives

Derivatives of Logarithmic Functions

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Wize University Calculus 1 Textbook > Derivatives

Logarithmic Differentiation

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Wize University Calculus 1 Textbook > Derivatives

Implicit Differentiation

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Wize University Calculus 1 Textbook > Derivatives

Differentiation Laws

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Compute the derivative of f(x)=xx+1f(x) = x^{x + 1}. Remember that log⁡x=log⁡ex=ln⁡x\log x = \log_e x = \ln x.
More Derivatives of Logarithmic Functions Questions:
Derivatives: Logarithmic Functions
Find the derivative of
f(x)=exx2+ln⁡(x)3f(x) = e^{x} x^2 + \frac{\ln(x)}{3}
Practice: Chain Rule
Find the derivative of y=ln⁡(arctan⁡x)y=\ln(\arctan x)
Derivatives: Exponential and Logarithmic Functions
Find f′(x)f'(x) if f(x)=ln⁡xxex\displaystyle f\left(x\right)=\frac{\ln x}{xe^{x}}. Simplify.
Inverse Trig with Log
Compute the derivative of f(x)=arctan(log10x)\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
Applying Rules of Logs
Find dydx\displaystyle\frac{dy}{dx} given y=log⁡6(x−1)5(x+1)103\displaystyle y=\log_6 \sqrt[3]{\frac{(x-1)^{5}}{(x+1)^{10}}}
Practice: Log and Ln
Q.\textbf{Q.} Find the derivative of y=log3(7xln⁡2)\displaystyle y=\text{log}_3(7x^{\ln2})
Derivatives of Logarithmic Functions: Two Logs
Find dy/dxdy/dx for y=5log⁡5(log⁡2t)\displaystyle y=5\log_{5}(\log_{2}t)
The chain rule: Logarithmic Derivatives
The derivative of log⁡3(e3x)\log_3\left(e^{3x}\right) is
Derivatives: Logarithmic Functions
Calculate the derivative of the following functions.
f(x)=ln⁡xx+xe2x\displaystyle f(x)=\frac{\ln{x}}{x+xe^{2x}}
Calculate the derivative of the following
f(x)=ln⁡xx+xe2x\displaystyle f(x)=\frac{\ln{x}}{x+xe^{2x}}
Derivatives: Logarithmic Functions
Find the derivative of the following function.
f(x)=ln⁡(x2+x)\displaystyle f(x)=\ln{(x^2+\sqrt{x})}
Derivatives: Logarithmic Functions
Compute the derivative of f(x)=log⁡2(x2+2xex)f(x)=\log_2(x^2+2xe^x)
Differentiate the following:
y=ex2+ln⁡xx\begin{aligned} &y=\frac{e^{x^2+\ln x}}{x} \end{aligned}
Derivatives: Exponential and Logarithmic Functions
Find f′(x)f'(x) if f(x)=ln⁡xxex\displaystyle f\left(x\right)=\frac{\ln x}{xe^{x}}. Simplify.
Inverse Trig with Log
Compute the derivative of f(x)=arctan(log10x)\displaystyle f(x)=\text{arctan}\left(\text{log}_{10}x\right)
Applying Rules of Logs
Find dydx\displaystyle\frac{dy}{dx} given y=log⁡6(x−1)5(x+1)103\displaystyle y=\log_6 \sqrt[3]{\frac{(x-1)^{5}}{(x+1)^{10}}}
Derivatives of Logarithmic Functions: Two Logs
Find dy/dxdy/dx for y=5log⁡5(log⁡2t)\displaystyle y=5\log_{5}(\log_{2}t)
Practice: Log and Ln
Q.\textbf{Q.} Find the derivative of y=log3(7xln⁡2)\displaystyle y=\text{log}_3(7x^{\ln2})
Derivatives: Logarithmic and Inverse Trigonometric Functions
Evaluate ddx(sin⁡−1xsin⁡xlog⁡3x)\displaystyle \frac{\text{d}}{\text{d}x}\left( \frac{\sin^{-1}x\sin x}{\log_3x}\right).
Derivatives: Trigonometric and Logarithmic Functions
Find ddy[(y5+ln⁡y)tan⁡y]\displaystyle \frac{\text{d}}{\text{d}y}\left[\left(y^5+\ln y\right)\tan y\right].
Compute the derivative of f(x)=log⁡2(x2+2xex)f(x) = \log_2(x^2 + 2xe^x).
Derivatives of Logarithmic Functions
Evaluate ddx(4xlog⁡700x)\frac{\text{d}}{\text{d}x}\left(4^x\log_{700}x\right). It is not necessary to simplify your answer.
Find the derivative of the following function.
f(x)=ln⁡(x2+x)\displaystyle f(x)=\ln{(x^2+\sqrt{x})}
Given that f(x)=ecos⁡(ln⁡x)f\left(x\right)=e^{\cos\left(\ln x\right)}, find f′(1)f'\left(1\right).
Practice: Chain Rule
Find the derivative of y=ln⁡(arctan⁡x)y=\ln(\arctan x)
Derivatives: Logarithmic Functions
Find the derivative of
f(x)=exx2+ln⁡(x)3f(x) = e^{x} x^2 + \frac{\ln(x)}{3}
Derivatives: Logarithmic Functions
Find all the points for which the tangent line of the function
f(x)=4ln⁡(x)+x2−3x+5f(x)= 4 \ln (x) + x^2 - 3x + 5
has slope equal to 3.
Derivatives of Logarithmic Functions: Normal Lines
Find the equation of the normal line to f(x)=ln⁡x at the point (e,1)f(x)=\ln{x} \text{ at the point } (e,1)
Note: This questions requires knowledge of derivatives of logarithms
Derivatives: Logarithmic Functions
Find the equation of the tangent line to the given point for the following function.
f(x)=ln⁡(x+3x+1)at x=1f(x)=\ln\left(\frac{x+3}{x+1}\right) \text{at}\ x=1
Derivatives: Exponential and Logarithmic Functions
Find the derivative of the following function.
f(x)=(x2+xln⁡x)exf(x)=(x^2+x\ln{x})e^{\sqrt{x}}
More Logarithmic Differentiation Questions:
Derivatives
Find the derivative of the following functions at the given values
a) f(x)=arcsin⁡(x2)x3−1f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1} at x=0x=0
b) g(x)=(cos⁡x)exg\left(x\right)=\left(\cos x\right)^{e^x} at x=0x=0
Logarithmic Differentiation
If f(x)=xxf(x)=x^{\sqrt{x}} then find f′(4)f'(4).
Implicit Differentiation: Logarithmic Differentiation
Find the equation of the tangent line to
(sin⁡(xy))x=x1/4(\sin(xy))^x = x^{1/4}
at the point (12,π2)\left(\dfrac{1}{2},\dfrac{\pi}{2}\right).
Logarithmic Differentiation
Use logarithmic differentiation to find g′g' where g(x)=xx22+tan⁡xg(x)=x^{x^2}\sqrt{2+\tan x}
Logarithmic Differentiation
If f(x)=(2x)sin⁡xf(x)=(2x)^{\sin x} , find f′(π2)f'(\frac{\pi}{2}) ?
Logarithmic Differentiation
If f(x)=xxf(x)=x^{\sqrt{x}} then find f′(4)f'(4).
Logarithmic Differentiation
Use logarithmic differentiation to find g′g' where g(x)=xx22+tan⁡xg(x)=x^{x^2}\sqrt{2+\tan x}
Complicated Log Derivative
If f(x)=(sin⁡x)3xf\left(x\right)=\left(\sin x\right)^{3x}, then f′(x)=f'\left(x\right)=
Logarithmic Differentiation: Big Product/Quotient
Consider y=(x2+1)2(x−1)4(x3+2)33\displaystyle y=\sqrt[3]{\frac{(x^{2}+1)^{2}(x-1)^{4}}{(x^{3}+2)^{3}}}. Compute dydx at x=0\displaystyle\frac{dy}{dx}\text{ at }x=0.
Logarithmic Differentiation
Find the derivative of y=(x+1)(x−1)y=\left(x+1\right)^{\left(x-1\right)}
Logarithmic differentiation
Find the derivative of the function
y=sin⁡(x)sin⁡(x)y = \sin(x)^{\sin(x)}
Logarithmic Differentiation
Find the derivative of y=(x+1)(x−1)y=\left(x+1\right)^{\left(x-1\right)}
Complicated Log Derivative
If f(x)=(sin⁡x)3xf\left(x\right)=\left(\sin x\right)^{3x}, then f′(x)=f'\left(x\right)=
Logarithmic Differentiation
Find the derivative of y=(x+1)(x−1)y=\left(x+1\right)^{\left(x-1\right)}
Logarithmic Differentiation
Find the derivative of y=(x+1)(x−1)y=\left(x+1\right)^{\left(x-1\right)}
Logarithmic differentiation
Find the derivative of the function
y=sin⁡(x)sin⁡(x)y = \sin(x)^{\sin(x)}
Logarithmic Differentiation
Find the derivative of y=(x+1)(x−1)y=\left(x+1\right)^{\left(x-1\right)}
Complicated Log Derivative
If f(x)=(sin⁡x)3xf\left(x\right)=\left(\sin x\right)^{3x}, then f′(x)=f'\left(x\right)=
Complicated Log Derivative
If f(x)=(sin⁡x)3xf\left(x\right)=\left(\sin x\right)^{3x}, then f′(x)=f'\left(x\right)=
Complicated Log Derivative
If f(x)=(sin⁡x)3xf\left(x\right)=\left(\sin x\right)^{3x}, then f′(x)=f'\left(x\right)=
Logarithmic Differentiation
Find the derivative of y=(x+1)(x−1)y=\left(x+1\right)^{\left(x-1\right)}
Logarithmic differentiation
Find the derivative of the function
y=sin⁡(x)sin⁡(x)y = \sin(x)^{\sin(x)}
Practice: Log Differentiation
Practice: Log Differentiation
Find the derivative of x2(cos⁡x)(e2x)2xln⁡(x)\frac{x^2\left(\cos x\right)\left(e^{2x}\right)}{2x\ln\left(x\right)}
Complicated Log Derivative
If f(x)=(sin⁡x)3xf\left(x\right)=\left(\sin x\right)^{3x}, then f′(x)=f'\left(x\right)=
Logarithmic Differentiation
Use logarithmic differentiation to find g′g' where g(x)=xx22+tan⁡xg(x)=x^{x^2}\sqrt{2+\tan x}
Logarithmic Differentiation
Find y′y' for the function 𝑦=𝑥2−x2𝑦 = 𝑥^{2-x^2}.
Find the derivative of y=(cos⁡x)5x2y=\left(\cos x\right)^{\frac{5x}{2}}
Logarithmic for Implicitly Defined Function
Find dy/dxdy/dx at the point (1,1)(1,1) given xy=yxx^y=y^x
Logarithmic Differentiation: Big Product/Quotient
Consider y=(x2+1)2(x−1)4(x3+2)33\displaystyle y=\sqrt[3]{\frac{(x^{2}+1)^{2}(x-1)^{4}}{(x^{3}+2)^{3}}}. Compute dydx at x=0\displaystyle\frac{dy}{dx}\text{ at }x=0.
Logarithmic with Chain
Use logarithmic differentiation to find g′g^\prime where g(x)=xx2+sin⁡x\displaystyle g(x)=x^x\sqrt{2+\sin{x}}.
Differentiation: Logarithmic with Trig
If y=sin⁡xxy=\sin^xx , find y′y'.
Derivatives
Find the derivative of the following functions at the given values
a) f(x)=arcsin⁡(x2)x3−1f\left(x\right)=\frac{\arcsin\left(x^2\right)}{x^3-1} at x=0x=0
b) g(x)=(cos⁡x)exg\left(x\right)=\left(\cos x\right)^{e^x} at x=0x=0
Logarithmic Differentiation
Find the derivative of y=(x+1)(x−1)y=\left(x+1\right)^{\left(x-1\right)}
Implicit Differentiation: Logarithmic Functions
For the following equations, solve for dydx\frac{dy}{dx} :
a) y=(5x2+3)sin⁡(x)y = (5x^2 + 3)^{\sin(x)}
b) x2y+3y2x2=x+yx^2y + 3y^2 x^2 = x + y
Logarithmic Differentiation
Calculate the derivative of the following functions.
f(x)=(tan⁡x)secx\displaystyle f(x)=(\tan{x})^{\text{sec}{x}}
g(x)=(xtan⁡4x)xg(x)=(x\tan{4x)}^x
Implicit Differentiation: Logarithmic Differentiation
Find the equation of the tangent line to
(sin⁡(xy))x=x1/4(\sin(xy))^x = x^{1/4}
at the point (12,π2)\left(\dfrac{1}{2},\dfrac{\pi}{2}\right).
Logarithmic Differentiation
Find the derivative of
f(x)=(tan⁡x)ln⁡xf(x)=(\tan x)^{\ln x}
Logarithmic Differentiation
Use logarithmic differentiation to find g' where
g(x)=xx2+sin⁡xg(x)=x^x\sqrt{2+\sin x}
Logarithmic Differentiation
Compute the derivative of
f(x)=x3/2x+1(x+5)7sin⁡xf(x)=\frac{x^{3/2}\sqrt{x+1}}{(x+5)^7\sin x}
Find the slope of the tangent line to g(x)=xx2+sin⁡xg(x)= x^x\sqrt{2+\sin x} when x=πx=\pi .
Differentiate the following functions.
(a) g(t)=(t5+t2)6tg(t)=(t^5+t^2)^{6t} (b) f(x)=x6xf(x)=\sqrt x^{6x} (c) f(x)=(x3+4)tan⁡xf(x)=(x^3+4)^{\tan x} (d) Finddydx\frac{dy}{dx} if ycos⁡(x)=xtan⁡(y)y\cos(x)=x\tan(y)
Calculate the derivative of the following function.
f(x)=(sin⁡x)cos⁡x\displaystyle f(x)=(\sin{x})^{\cos{x}}
Logarithmic Differentiation
Find y′y' for the function 𝑦=𝑥2−x2𝑦 = 𝑥^{2-x^2}.
Logarithmic Differentiation
Use logarithmic differentiation to find g′g' where g(x)=xx22+tan⁡xg(x)=x^{x^2}\sqrt{2+\tan x}
Compute the derivative of f(x)=x3/2x+1(x+5)7sin⁡xf(x) = \dfrac{x^{3/2}\sqrt{x+1}}{(x+5)^7\sin x}
Find the derivative of y=(cos⁡x)5x2y=\left(\cos x\right)^{\frac{5x}{2}}
Logarithmic Differentiation: Big Product/Quotient
Consider y=(x2+1)2(x−1)4(x3+2)33\displaystyle y=\sqrt[3]{\frac{(x^{2}+1)^{2}(x-1)^{4}}{(x^{3}+2)^{3}}}. Compute dydx at x=0\displaystyle\frac{dy}{dx}\text{ at }x=0.
Logarithmic for Implicitly Defined Function
Find dy/dxdy/dx at the point (1,1)(1,1) given xy=yxx^y=y^x
Logarithmic with Chain
Use logarithmic differentiation to find g′g^\prime where g(x)=xx2+sin⁡x\displaystyle g(x)=x^x\sqrt{2+\sin{x}}.
Differentiation: Logarithmic with Trig
If y=sin⁡xxy=\sin^xx , find y′y'.
Logarithmic Differentiation
Consider y=(x2+1)2(x−1)4(x3+2)33\displaystyle y=\sqrt[3]{\frac{(x^{2}+1)^{2}(x-1)^{4}}{(x^{3}+2)^{3}}}. Compute dydx at x=0\displaystyle\frac{dy}{dx}\text{ at }x=0.
Logarithmic differentiation
Find the derivative of f(x)=(x+1)9ex(sin⁡ x)32x5x2+1f\left(x\right)=\frac{\left(x+1\right)^9e^x\left(\sin\ x\right)^{\frac{3}{2}}}{x^5\sqrt{x^2+1}}.
Logarithmic Differentiation
Find the derivative of y=(x−3)9 xx ln⁡ xx−9.y=\frac{\left(x-3\right)^9\ x^x\ \ln\ x}{x-9}.
Logarithmic Differentiation
If f(x)=(2x)sin⁡xf(x)=(2x)^{\sin x} , find f′(π2)f'(\frac{\pi}{2}) ?
Logarithmic Differentiation
If f(x)=xxf(x)=x^{\sqrt{x}} then find f′(4)f'(4).
Find the derivative of f(x)=(tan⁡x)ln⁡ x.f(x)=(\tan x)^{\ln\ x}.
Find the derivative of f(x)=(tan⁡x)ln⁡xf(x)=(\tan x)^{\ln x}.
Calculate the derivative of the following functions.
f(x)=(tan⁡x)secx\displaystyle f(x)=(\tan{x})^{\text{sec}{x}}
Find f′(1)f'(1) if f(x)=(arctan⁡x)x2f\left(x\right)=\left(\arctan x\right)^{x^2}.
Practice: Log Differentiation
Practice: Log Differentiation
Find the derivative of x2(cos⁡x)(e2x)2xln⁡(x)\frac{x^2\left(\cos x\right)\left(e^{2x}\right)}{2x\ln\left(x\right)}
Complicated Log Derivative
If f(x)=(sin⁡x)3xf\left(x\right)=\left(\sin x\right)^{3x}, then f′(x)=f'\left(x\right)=
Logarithmic differentiation
Find the derivative of the function
y=sin⁡(x)sin⁡(x)y = \sin(x)^{\sin(x)}
Logarithmic Differentiation
Calculate the derivative of the following functions.
f(x)=(sin⁡x)cos⁡x\displaystyle f(x)=(\sin{x})^{\cos{x}}
Logarithmic Differentiation
Calculate the derivative of the following functions.
f(x)=xx2+sin⁡x\displaystyle f(x)=x^x\sqrt{2+\sin{x}}
Logarithmic Differentiation
Calculate the derivative of the following functions.
f(x)=(tan⁡x)secx\displaystyle f(x)=(\tan{x})^{\text{sec}{x}}
Practice: Logarithmic Differentiation
Find the derivative of f(x)=sin⁡xcos⁡xf(x)=\sin{x}^{\cos{x}}
More Implicit Differentiation Questions:
Implicit differentiation
Differentiate exy−tan⁡y=3x3+2xe^{xy}-\tan y=\frac{3}{x^3}+2x
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
Tangent lines: Implicit Differentiation
An equation of the tangent line to the curve x3+xln⁡y+y4=3x32+exx^3+x\ln y+y^4=3x^{\frac{3}{2}}+e^x at x=0x=0 is
Implicit Differentiation: Logarithmic Differentiation
Find the equation of the tangent line to
(sin⁡(xy))x=x1/4(\sin(xy))^x = x^{1/4}
at the point (12,π2)\left(\dfrac{1}{2},\dfrac{\pi}{2}\right).
Implicit Second Derivative
Find d2ydx2\dfrac{d^2y}{dx^2} at the point (2,2)(2,2) given xy+y2=2xy+y^2=2
Implicit differentiation
Differentiate exy−tan⁡y=3x3+2xe^{xy}-\tan y=\frac{3}{x^3}+2x
Implicit Differentiation
Find the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point (1,9).
Tangent lines: Implicit Differentiation
An equation of the tangent line to the curve x3+xln⁡y+y4=3x32+exx^3+x\ln y+y^4=3x^{\frac{3}{2}}+e^x at x=0x=0 is
Implicit Differentiation
Find y'' if x3+y3=4.x^3+y^3=4.
Implicit Differentiation
Find dydx\displaystyle \frac{\text{d}y}{\text{d}x} for y3=2xy^3=2x.
Implicit Differentiation
Find the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point (1,9).
Implicit differentiation
Find the derivative of the following:
yex+y2=x2ye^{x+y^2}=x^2
Tangent lines: Implicit Differentiation
An equation of the tangent line to the curve x3+xln⁡y+y4=3x32+exx^3+x\ln y+y^4=3x^{\frac{3}{2}}+e^x at x=0x=0 is
Find the tangent to the curve 4𝑦2 − 𝑒𝑥 + 𝑥 = 4𝑥 + 3 at the point where 𝑥 = 0 and 𝑦 is positive
Implicit Differentiation
Evaluate the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point P(1,9)P(1,9).
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
Implicit Differentiation
Evaluate the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point P(1,9)P(1,9).
Implicit Differentiation
For the implicit equation x4+x3y−xy2=1x^4+x^3y-xy^2=1, where y = f(x), find y'.
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
Implicit Differentiation
Find all the equations of the tangent lines to the curve y=x3+x2−xy=x^3+x^2-x that are parallel to the x-axis.
Find dydx\frac{dy}{dx} at x=0x=0 if xcos⁡(y2)=ycos⁡xx\cos\left(\frac{y}{2}\right)=y\cos x.
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
Implicit differentiation
Differentiate exy−tan⁡y=3x3+2xe^{xy}-\tan y=\frac{3}{x^3}+2x
Implicit Differentiation
Given x2+y=y2−4x^2+y=y^2-4, find y′y'.
Implicit Differentiation
Find dydx\displaystyle \frac{\text{d}y}{\text{d}x} for y3=2xy^3=2x.
Find the point(s) where x2−4x+6y+y2=−9 x^2-4x+6y+y^2 = −9 has horizontal and vertical tangent lines.
Find dydx\dfrac{dy}{dx} for the function ycos⁡x=x2+cos⁡yy\cos x=x^2+\cos y
Logarithmic for Implicitly Defined Function
Find dy/dxdy/dx at the point (1,1)(1,1) given xy=yxx^y=y^x
Practice: Tangent and Normal Lines with Implicit
Q:\textbf{Q:} Find the equation of the tangent and the normal lines to the curve x2cos⁡2y−sin⁡y=0x^2\cos^2y-\sin y=0 at the point (0,π)(0,\pi).
Implicit Second Derivative
Find d2ydx2\dfrac{d^2y}{dx^2} at the point (2,2)(2,2) given xy+y2=2xy+y^2=2
Implicit Differentiation
Evaluate the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point P(1,9)P(1,9).
Implicit with Trig
Find dy/dxdy/dx for xy=tan⁡(x+y2)\displaystyle xy=\tan(x+y^{2})
Implicit Differentiation
Let xx, yy satisfy the equation xy=6y−x+1x^y = 6y - x + 1. Find dydx\frac{dy}{dx} at the point (x,y)=(1,1/6)(x, y) = (1, 1/6).
Tangent lines: Implicit Differentiation
An equation of the tangent line to the curve x3+xln⁡y+y4=3x32+exx^3+x\ln y+y^4=3x^{\frac{3}{2}}+e^x at x=0x=0 is
Implicit Differentiation: Logarithmic Functions
For the following equations, solve for dydx\frac{dy}{dx} :
a) y=(5x2+3)sin⁡(x)y = (5x^2 + 3)^{\sin(x)}
b) x2y+3y2x2=x+yx^2y + 3y^2 x^2 = x + y
Evaluate the slope of the tangent line to the curve at the given point.
xy=x3y−6  at  P(1,9)\displaystyle \sqrt{xy}=x^3y-6\,\, at \,\,P(1,9)
Implicit differentiation
Find the derivative of the following:
yex+y2=x2ye^{x+y^2}=x^2
Implicit differentiation
Find the derivative of the following:
sin⁡(x+y)−sec⁡(x2+y2)=y\sin(x+y)-\mathrm{\sec}(x^2+y^2)=y
Implicit Differentiation
Evaluate the slope of the tangent line to the curve at the given point.
x3+y3=ysin⁡x+1,  at  P(0,1)x^3+y^3=y\sin{x}+1,\,\, \text{at}\,\,P(0,1)
Evaluate the slope of the tangent line to the curve at the given point.
xy=x3y−6  at  P(1,9)\displaystyle \sqrt{xy}=x^3y-6\,\, at \,\,P(1,9)
Evaluate the slope of the tangent line to the curve at the given point.
xy=x3y−6  at  P(1,9)\displaystyle \sqrt{xy}=x^3y-6\,\, at \,\,P(1,9)
Find the equation of the tangent line to the equation x3+ln⁡y=8x^3+\ln y=8 at the point (2, 1)
Evaluate the slope of the tangent line to the curve at the given point.
xy=x3y−6  at  P(1,9)\displaystyle \sqrt{xy}=x^3y-6\,\, at \,\,P(1,9)
Implicit Differentiation: Logarithmic Differentiation
Find the equation of the tangent line to
(sin⁡(xy))x=x1/4(\sin(xy))^x = x^{1/4}
at the point (12,π2)\left(\dfrac{1}{2},\dfrac{\pi}{2}\right).
Implicit Differentiation
Find the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point (1,9).
Implicit Differentiation
Find y'' if x3+y3=4.x^3+y^3=4.
Find dydx\frac{dy}{dx} given that exy−xy=2x2+1 e^{xy}-xy=2x^2+1
final114
Find y′y' for the function x4+x3y−xy2=1x^4+x^3y-xy^2=1
Differentiate the following functions.
(a) g(t)=(t5+t2)6tg(t)=(t^5+t^2)^{6t} (b) f(x)=x6xf(x)=\sqrt x^{6x} (c) f(x)=(x3+4)tan⁡xf(x)=(x^3+4)^{\tan x} (d) Finddydx\frac{dy}{dx} if ycos⁡(x)=xtan⁡(y)y\cos(x)=x\tan(y)
Differentiate y with respect to x in the equation 2x2+3xy−y2=12x^2+3xy-y^2=1.
Find dydx\frac{dy}{dx} given that exy−xy=2x2+1 e^{xy}-xy=2x^2+1
Evaluate the slope of the tangent line to the curve at the given point.
x3+y3=4xy,  at  P(2,2)x^3+y^3=4xy ,\,\, \text{at} \,\, P(2,2)
Evaluate the slope of the tangent line to the curve at the given point.
x3+y3=ysin⁡x+1,  at  P(0,1)x^3+y^3=y\sin{x}+1,\,\, \text{at}\,\,P(0,1)
Implicit Differentiation
Given x2+y=y2−4x^2+y=y^2-4, find y′y'.
Implicit Differentiation
Find dydx\displaystyle \frac{\text{d}y}{\text{d}x} for y3=2xy^3=2x.
Find the point(s) where x2−4x+6y+y2=−9 x^2-4x+6y+y^2 = −9 has horizontal and vertical tangent lines.
Find dydx\dfrac{dy}{dx} for the function ycos⁡x=x2+cos⁡yy\cos x=x^2+\cos y
Logarithmic for Implicitly Defined Function
Find dy/dxdy/dx at the point (1,1)(1,1) given xy=yxx^y=y^x
Implicit Second Derivative
Find d2ydx2\dfrac{d^2y}{dx^2} at the point (2,2)(2,2) given xy+y2=2xy+y^2=2
Practice: Tangent and Normal Lines with Implicit
Q:\textbf{Q:} Find the equation of the tangent and the normal lines to the curve x2cos⁡2y−sin⁡y=0x^2\cos^2y-\sin y=0 at the point (0,π)(0,\pi).
Implicit Differentiation
Evaluate the slope of the line tangent to the curve xy=x3y−6\sqrt{xy}=x^3y-6 at the point P(1,9)P(1,9).
Implicit with Trig
Find dy/dxdy/dx for xy=tan⁡(x+y2)\displaystyle xy=\tan(x+y^{2})
Find the slope of the tangent line to the curve ln⁡(x−y)+3y2=3\ln(x-y)+3y^2=3 at point (3,2)(3,2).
Find y′y' for the function x4+x3y−xy2=1x^4+x^3y-xy^2=1
Implicit Differentiation: Estimate of Implicit Function
Use a linear approximation to estimate the yy value of the curve described by x3+3xy+y3=5x^3+3xy+y^3=5 at the point for which x=0.1x=0.1 (leave it as an exact value if you're not allowed a calculator on the exam).
Find the point on the graph y2 = 4x + 5 where the tangent line is parallel to the line y = 2x + 184.
Concept Clarifier
Compute the second derivative of the implicit function y3=xcos⁡yy^3=x \cos y, where yy is a function of xx. Do not simplify.
Find the point(s) where x2−4x+6y+y2 = −9 has horizontal and vertical tangent lines.
Find the slope of the tangent line to the curve ln(x − y) + y2 = 4 at point (3,2).
(Express your answer as a fraction in lowest terms. If the answer is a negative number, put the negative sign in front of the entire fraction.)
Find the equation of the tangent line at the point (1,2)(1,2) of the curve y3−2x5y2=yln⁡(x)y^3-2x^5y^2=y\ln(x).
Find the derivative of sin⁡(x+y)−sec(x2+y2)=y\sin(x+y)-\mathrm{sec} (x^2+y^2)=y.
Find the derivative of y with respect to x of the following equation, y2+x5y3=5−ln⁡yy^2+x^5y^3=5-\ln y.
Given that x2+y=y2−4x^2+y=y^2-4, where y is dependent on x, find dydx\frac{dy}{dx}.
Implicit Differentiation
For the implicit equation x4+x3y−xy2=1x^4+x^3y-xy^2=1, where y = f(x), find y'.
Evaluate the slope of the tangent line to the curve at the given point.
xy=x3y−6  at  P(1,9)\displaystyle \sqrt{xy}=x^3y-6\,\, at \,\,P(1,9)
Find the tangent to the curve 4𝑦2 − 𝑒𝑥 + 𝑥 = 4𝑥 + 3 at the point where 𝑥 = 0 and 𝑦 is positive
Implicit Differentiation
Find all the equations of the tangent lines to the curve y=x3+x2−xy=x^3+x^2-x that are parallel to the x-axis.
Find dydx\frac{dy}{dx} at x=0x=0 if xcos⁡(y2)=ycos⁡xx\cos\left(\frac{y}{2}\right)=y\cos x.
Implicit Differentiation: Tangent line
Find the equation of the tangent to the graph 3y3+x=x2+13y^3+x=x^2+1 at x=2x=2.
Implicit differentiation
Differentiate exy−tan⁡y=3x3+2xe^{xy}-\tan y=\frac{3}{x^3}+2x
Implicit differentiation
Suppose 3x3+4y2=193x^3+4y^2=19, where x, yx,\ y are both functions of tt.
If dydt=3\frac{dy}{dt}=3, find dxdt\frac{dx}{dt} when y=2y=2.
Solve for dydx\frac{dy}{dx} :
x2y+3y2x2=x+yx^2y+3y^2x^2=x+y
Implicit Differentiation
Use implicit differentiation to calculate the derivative of
θ(x)=arctan⁡(x)\theta(x) = \arctan(x)
Implicit Differentiation
Find dydx\displaystyle \frac{\text{d}y}{\text{d}x} for ln⁡(x2+y2)+2xy=4\ln(x^2+y^2)+2xy=4.
Implicit Differentiation
Evaluate the slope of the tangent line to the curve at the given point.
x3y2+xtan⁡y=4 at P(x,y)\displaystyle x^3y^2+x\tan{y}=4 \, at\, P(x,y)
Evaluate the slope of the tangent line to the curve at the given point.
x2y3+xln⁡y+ysin⁡x=10  at  P(x,y)\displaystyle x^2y^3+x\ln{y}+y\sin{x}=10\,\, at \,\,P(x,y)
Practice: Implicit Differentiation
Find dydx\frac{dy}{dx} for the following
tan⁡yx=x\tan{\frac{y}{x}}=x
More Differentiation Laws Questions:
Derivative of Polynomials: Product Rule
Find the derivative of f(x)=(x3/2+2x)(x7/4+x)f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
Differentiation laws
What is the derivative of the function f(x)=x3+1x2f(x)=\frac{x^3+1}{x^2}?
Differentiation: Piecewise Differentiable Function
Find AA and BB for which f(x)f\left(x\right) is differentiable everywhere. Answers are in the form (A,B)\left(A,B\right)
f(x)={xex2+1, if x≥1Ax+B, if x<1f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
Differentiation: Piecewise Differentiable Function
Find AA and BB for which f(x)f\left(x\right) is differentiable everywhere.
f(x)={xex2+1, if x≥1Ax+B, if x<1f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
Derivative of Polynomials: Product Rule
Find the derivative of f(x)=(x3/2+2x)(x7/4+x)f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
Differentiation: Piecewise Differentiable Function
Find AA and BB for which f(x)f\left(x\right) is differentiable everywhere.
f(x)={xex2+1, if x≥1Ax+B, if x<1f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
Differentiation: Piecewise Differentiable Function
Find AA and BB for which f(x)f\left(x\right) is differentiable everywhere.
f(x)={xex2+1, if x≥1Ax+B, if x<1f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
Differentiation: Piecewise Differentiable Function
Find AA and BB for which f(x)f\left(x\right) is differentiable everywhere.
f(x)={xex2+1, if x≥1Ax+B, if x<1f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
The chain rule
Find the derivative of h(x)=log⁡(cos⁡(x))h(x) = \log(\cos(x)). Remember that log⁡x=log⁡ex=ln⁡x\log x = \log_e x = \ln x.
The Quotient Rule
Find the derivative of g(x)=x2−52x+1g(x) = \frac{x^2 - 5}{2x + 1}
The product rule
Find the derivative of f(x)=x3exf(x) = x^3 e^x
Differentiation: Piecewise Differentiable Function
Find AA and BB for which f(x)f\left(x\right) is differentiable everywhere.
f(x)={xex2+1, if x≥1Ax+B, if x<1f(x)=\begin{cases} xe^{x^2+1}, \text{ if } x\geq 1 \\ Ax+B, \text{ if } x < 1 \end{cases}
Derivative of Polynomials: Product Rule
Find the derivative of f(x)=(x3/2+2x)(x7/4+x)f(x)=\left(x^{3/2}+\frac{2}{x}\right)\left(x^{7/4}+x\right)
Differentiation laws
What is the derivative of the function f(x)=x3+1x2f(x)=\frac{x^3+1}{x^2}?
Is f(x)f(x) differentiable at x=1x=1 ? If so, find f′(1)f'(1).
f(x)={x+1if x<112x2+32if x≥1 f(x) = \begin{cases} x+1 & \text{if } x < 1 \\ \frac{1}{2}x^2 + \frac{3}{2} & \text{if } x \geq 1 \end{cases}
Differential Laws: nth Derivatives
If y=(10e+1)10y=\left(10e+1\right)^{10}, find the 9th derivative of yy.
If f(x)f(x) is differentiable everywhere, find AA and BB.
f(x)={x2+1     if x≥0Ax+B       if  x<0f(x)=\left\{ \begin{array}{ll} \displaystyle x^2+1\quad\quad\,\,\,\,\,\text{if}\,x\geq 0 \\ Ax+B\quad\,\,\,\,\,\,\,\text{if}\,\, x<0\\ \end{array} \right.