Practice Implicit Differentiation in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Finding $\frac{dy}{dx}$ when $y$ is defined implicitly by an equation like $F(x, y) = 0$ , by differentiating both sides and solving for $\frac{dy}{dx}$ .

Sometimes you can't (or don't want to) solve for $y$ explicitly. Instead, differentiate the whole equation as-is. Every time you differentiate a $y$ -term, attach $\frac{dy}{dx}$ by the chain rule (since $y$ secretly depends on $x$ ), then solve for $\frac{dy}{dx}$ .

Showing a random 20 of 50 problems.

Example 1

easy

Differentiate

y = x^2

both implicitly and explicitly; confirm they agree.

Example 2

easy

Find

\frac{dy}{dx}

for

4x + 5y = 20

Example 3

medium

Find

\frac{dy}{dx}

for

\tan(y) = x

Example 4

easy

Find

\frac{d}{dx}[x^2 y]

where

y = y(x)

Example 5

hard

Find the equation of the normal line to

x^2 + xy + y^2 = 3

at the point

(1, 1)

Example 6

easy

Find

\frac{dy}{dx}

for the circle

x^2 + y^2 = 25

and evaluate it at the point

(3, 4)

Example 7

easy

What does the chain rule contribute when differentiating

\sin y

with respect to

x

Example 8

medium

Find

\frac{dy}{dx}

for

e^y = x + y

Example 9

easy

Find

\frac{dy}{dx}

for

x^2 + 4y^2 = 16

Example 10

medium

Find

\frac{dy}{dx}

for

\sqrt{x} + \sqrt{y} = 4

Example 11

medium

Find

\frac{dy}{dx}

for

x^2 + xy + y^2 = 7

Example 12

medium

Find

\frac{dy}{dx}

for

\sin x + \cos y = 1

Example 13

medium

Find

\frac{dy}{dx}

for

x^3 + y^3 = 6xy

(the folium of Descartes).

Example 14

hard

Find

\frac{dy}{dx}

for

y = x^x

(use logarithmic differentiation).

Example 15

medium

Find

\frac{dy}{dx}

for the circle

x^2 + y^2 = 25

Example 16

easy

For

y^2 = x

, find

\frac{dy}{dx}

implicitly.

Example 17

easy

Find

\frac{d}{dx}[xy]

Example 18

easy

Differentiate

x^2 + y^2 = 25

implicitly to find

\frac{dy}{dx}

Example 19

easy

Differentiate

x^2 + y = 7

for

\frac{dy}{dx}

Example 20

challenge

Find the tangent line to

x^{2/3} + y^{2/3} = 4

(astroid) at

(1, 3\sqrt{3})

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Related Concepts

Derivative Chain Rule Differentiation Rules Related Rates