Implicit Differentiation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Implicit Differentiation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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 $\frac{dy}{dx}$ .

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: When you can't isolate $y$ , differentiate the whole equation and attach $\frac{dy}{dx}$ to each $y$ -term via the chain rule, then solve.

Common stuck point: The procedure for implicit differentiation is the easy part; the trap is differentiating a $y$ -term without attaching $\frac{dy}{dx}$ . Asking "Is $y$ tied to $x$ by an equation I can't easily solve for $y$ , and do I need its derivative?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is $y$ tied to $x$ by an equation I can't easily solve for $y$ , and do I need its derivative?

Worked Examples

Example 1

easy

Find

\frac{dy}{dx}

for the circle

x^2 + y^2 = 25

and evaluate it at the point

(3, 4)

Answer

\dfrac{dy}{dx} = -\dfrac{x}{y}

; at

(3, 4)

: slope

= -\dfrac{3}{4}

First step

Differentiate both sides with respect to

x

2x + 2y\frac{dy}{dx} = 0

Full solution

2
Solve for $\frac{dy}{dx}$ : $\frac{dy}{dx} = -\frac{x}{y}$ .
3
At $(3, 4)$ : $\frac{dy}{dx} = -\frac{3}{4}$ .

Whenever a

y

-term is differentiated, attach

\frac{dy}{dx}

by the chain rule. Then collect all

\frac{dy}{dx}

terms and solve. The tangent to a circle at

(3,4)

has slope

-3/4

Example 2

hard

Find

\frac{dy}{dx}

for

x^3 + y^3 = 6xy

(folium of Descartes).

Example 3

medium

Find

\frac{dy}{dx}

for the circle

x^2 + y^2 = 25

Example 4

medium

Find

\frac{dy}{dx}

for

x^2 y + xy^2 = 6

Example 5

medium

Find where the tangent to the ellipse

4x^2 + 9y^2 = 36

is vertical.

Example 6

hard

A ladder 10 ft long leans against a wall. If the bottom slides away from the wall at 2 ft/s, how fast is the top sliding down when the bottom is 6 ft from the wall?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find

\frac{dy}{dx}

for

x^2 + 3y^2 = 7

Example 2

medium

Find the equation of the tangent line to

x^2y + y^3 = 2

at the point

(1, 1)

Example 3

easy

Differentiate

x^2 + y^2 = 25

implicitly to find

\frac{dy}{dx}

Example 4

easy

Find

\frac{d}{dx}[y^3]

treating

y

as a function of

x

Example 5

easy

Find

\frac{d}{dx}[xy]

Example 6

easy

Differentiate

y = x^2

both implicitly and explicitly; confirm they agree.

Example 7

easy

Find

\frac{dy}{dx}

for

3x + 2y = 12

Example 8

easy

Differentiate

x^2 + y = 7

for

\frac{dy}{dx}

Example 9

easy

What does the chain rule contribute when differentiating

\sin y

with respect to

x

Example 10

easy

For

y^2 = x

, find

\frac{dy}{dx}

implicitly.

Example 11

medium

Find

\frac{dy}{dx}

for

x^2 + xy + y^2 = 7

Example 12

medium

Find the slope of

x^2 + y^2 = 25

at the point

(3, 4)

Example 13

medium

Find

\frac{dy}{dx}

for

\sin(xy) = x

Example 14

medium

Find the equation of the tangent to

x^2 + y^2 = 25

(3, 4)

Example 15

medium

Find

\frac{dy}{dx}

for

e^y = x + y

Example 16

medium

Find

\frac{dy}{dx}

for

x^3 + y^3 = 6xy

(the folium of Descartes).

Example 17

medium

Find where the tangent to

x^2 + y^2 = 25

is horizontal.

Example 18

medium

Find

\frac{dy}{dx}

for

\ln(y) + xy = 3

Example 19

medium

Find

\frac{d^2y}{dx^2}

for

x^2 + y^2 = 25

Example 20

challenge

Find the tangent line to

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

(astroid) at

(1, 3\sqrt{3})

Example 21

challenge

For

xy + y^2 = 6

, find

\frac{dy}{dx}

at the point(s) where

x = 1

Example 22

challenge

Show that for

xy = c

(constant

c>0

), the tangent at any point cuts off a triangle of constant area

2c

with the axes.

Example 23

easy

Find

\frac{dy}{dx}

for

4x + 5y = 20

Example 24

easy

Find

\frac{dy}{dx}

for

x^2 + 4y^2 = 16

Example 25

easy

Find

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

where

y = y(x)

Example 26

easy

Find

\frac{dy}{dx}

for

y^3 = x + 1

Example 27

easy

Find

\frac{dy}{dx}

for

x + y^2 = 9

Example 28

medium

Find

\frac{dy}{dx}

for

\sin x + \cos y = 1

Example 29

medium

Find

\frac{dy}{dx}

for

e^{xy} = x + y

Example 30

medium

Find the slope of the tangent to

x^2 - xy + y^2 = 7

(3, 2)

Example 31

medium

Find the equation of the tangent to

x^2 + y^2 = 25

(-4, 3)

Example 32

medium

Find

\frac{dy}{dx}

for

\tan(y) = x

Example 33

medium

Find

\frac{dy}{dx}

for

x^2 y + \sin y = 4

Example 34

medium

Find

\frac{dy}{dx}

for

\ln(xy) = x + y

Example 35

hard

Find

\frac{d^2 y}{dx^2}

for

x^2 + y^2 = 1

at the point

(\tfrac{1}{2}, \tfrac{\sqrt{3}}{2})

Example 36

hard

Find all points on

x^2 + xy + y^2 = 3

where the tangent line is horizontal.

Example 37

hard

Find

\frac{dy}{dx}

for

y = x^x

(use logarithmic differentiation).

Example 38

hard

Find

\frac{dy}{dx}

for

\arcsin(y) + xy = 1

Example 39

hard

Find the equation of the normal line to

x^2 + xy + y^2 = 3

at the point

(1, 1)

Example 40

challenge

For the curve

y^2 = x^3 - x

(an elliptic curve), find all points where the tangent is horizontal.

Example 41

challenge

Show that for the curve

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

(astroid) the portion of any tangent line cut off by the axes has constant length

a

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Related Concepts

Derivative Chain Rule Differentiation Rules Related Rates

Background Knowledge

These ideas may be useful before you work through the harder examples.

derivativechain ruledifferentiation rules