Implicit differentiation

A circle $x^{2} + y^{2} = 25$ isn't one function, but it still has a tangent at each point. Treat $y$ as a function of $x$ and use the chain rule on it.

x^{2} + y^{2} = 25

(3, 4)

. The slope of the tangent is

- \frac{3}{4}

Work through it

Find $\frac{d y}{d x}$ for $x^{2} + y^{2} = 25$ .

Differentiate both sides. $2 x + 2 y \frac{d y}{d x} = 0$

Solve. $\frac{d y}{d x} = - \frac{x}{y}$

At $(3, 4)$ . $- \frac{3}{4}$

Find $\frac{d y}{d x}$ for $x y = 6$ .

Use the product rule on the left. $y + x \frac{d y}{d x} = 0$

Solve. $\frac{d y}{d x} = - \frac{y}{x}$

Watch out for this

Differentiating $y^{2}$ as $2 y$ .

$y$ depends on $x$ , so the chain rule gives $2 y \frac{d y}{d x}$ .

The short version
Differentiate both sides with respect to

x

. Any

y

term picks up a

\frac{d y}{d x}

. Then solve for it.

Your turn

Prefer paper? Print the More derivatives and applications worksheet, with an answer key.

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Work through it

Find dxdy​ for x2+y2=25.

Find dxdy​ for xy=6.

Watch out for this

Your turn

Find $\frac{d y}{d x}$ for $x^{2} + y^{2} = 25$ .

Find $\frac{d y}{d x}$ for $x y = 6$ .