3.2 Implicit Differentiation

Learning objectives

—FUN-3.D—Calculate derivatives of implicitly defined functions• FUN-3.D Calculate derivatives of implicitly defined functions.• FUN-3.D.1 The chain rule is the basis for implicit differentiation.• Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

An implicit equation links $x$ and $y$ without necessarily isolating $y$ . Treat $y$ as a differentiable function $y(x)$ . When a term involving $y$ is differentiated with respect to $x$ , the chain rule introduces a factor of $dy/dx$ .

\frac{d}{dx}[F(y)]=F'(y)\frac{dy}{dx}

Differentiate every term on both sides with respect to $x$ .
Attach $dy/dx$ whenever the chain rule differentiates a function of $y$ .
Move all terms containing $dy/dx$ to one side.
Factor out $dy/dx$ and solve for it.

For $x^2+y^2=25$ , differentiate with respect to $x$ : $2x+2y\frac{dy}{dx}=0.$ Therefore $2y\frac{dy}{dx}=-2x\quad\Rightarrow\quad\frac{dy}{dx}=-\frac{x}{y}.$ At $(3,4)$ , the slope is $-3/4$ .

Do not differentiate $y^2$ as merely $2y$ : because $y$ depends on $x$ , the result is $2y\,dy/dx$ . The solved formula $-x/y$ applies where $y\ne 0$ ; when $y=0$ on this circle, the tangent is vertical rather than having a finite $dy/dx$ .

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—3.3 Differentiating Inverse FunctionsRead