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Implicit Differentiation: Estimate of Implicit Function

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Wize University Calculus 1 Textbook > Derivatives

Implicit Differentiation

4 Activities

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Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
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  3. View the solution and report whether you got it right or wrong.
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).



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).
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.
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
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