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Wize University Calculus 2 Textbook > Review: Derivatives

Implicit Differentiation

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Implicit Differentiation

Sometimes, you will be asked to differentiate an expression that is not given in the form y=...y=....
  • There may be multiple yy terms, or
  • You may not be able to easily solve for y=...y=...
Examples:
  • 3cos⁡(3x)+4y3=yex−ln⁡x3\cos\left(3x\right)+4y^3=ye^x-\ln x

Steps

  • Differentiate the equation term-by-term as usual, one side at a time
  • When you differentiate a 𝑦 term, don’t forget to multiply by a 𝑑𝑦/𝑑𝑥
  • Solve for 𝑑𝑦/𝑑𝑥 in the end
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Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.

Differentiate exy−tan⁡y=3x3+2xe^{xy}-\tan y=\frac{3}{x^3}+2x








Extra Practice
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 Second Derivative
Find d2ydx2\dfrac{d^2y}{dx^2} at the point (2,2)(2,2) given xy+y2=2xy+y^2=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
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
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
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).
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 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 Differentiation
For the implicit equation x4+x3y−xy2=1x^4+x^3y-xy^2=1, where y = f(x), find y'.
Implicit Differentiation
Use implicit differentiation to calculate the derivative of
θ(x)=arctan⁡(x)\theta(x) = \arctan(x)