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g(x)=(xtan4x)^x

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Wize University Calculus 1 Textbook > Derivatives

Logarithmic Differentiation

3 Activities


g(x)=(xtan⁡4x)xg(x)=(x\tan{4x)}^x


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Using the limit definition of the derivative, write an explicit formula for g′(x)g'(x)

Do not attempt to simplify or evaluate the limit.
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: Logarithmic Functions
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.
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}}
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}}