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

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

Showing a random 20 of 50 problems.

Example 1

easy
Differentiate y=x2y = x^2 both implicitly and explicitly; confirm they agree.

Example 2

easy
Find dydx\frac{dy}{dx} for 4x+5y=204x + 5y = 20.

Example 3

medium
Find dydx\frac{dy}{dx} for tan⁡(y)=x\tan(y) = x.

Example 4

easy
Find ddx[x2y]\frac{d}{dx}[x^2 y] where y=y(x)y = y(x).

Example 5

hard
Find the equation of the normal line to x2+xy+y2=3x^2 + xy + y^2 = 3 at the point (1,1)(1, 1).

Example 6

easy
Find dydx\frac{dy}{dx} for the circle x2+y2=25x^2 + y^2 = 25 and evaluate it at the point (3,4)(3, 4).

Example 7

easy
What does the chain rule contribute when differentiating sin⁡y\sin y with respect to xx?

Example 8

medium
Find dydx\frac{dy}{dx} for ey=x+ye^y = x + y.

Example 9

easy
Find dydx\frac{dy}{dx} for x2+4y2=16x^2 + 4y^2 = 16.

Example 10

medium
Find dydx\frac{dy}{dx} for x+y=4\sqrt{x} + \sqrt{y} = 4.

Example 11

medium
Find dydx\frac{dy}{dx} for x2+xy+y2=7x^2 + xy + y^2 = 7.

Example 12

medium
Find dydx\frac{dy}{dx} for sin⁡x+cos⁡y=1\sin x + \cos y = 1.

Example 13

medium
Find dydx\frac{dy}{dx} for x3+y3=6xyx^3 + y^3 = 6xy (the folium of Descartes).

Example 14

hard
Find dydx\frac{dy}{dx} for y=xxy = x^x (use logarithmic differentiation).

Example 15

medium
Find dydx\frac{dy}{dx} for the circle x2+y2=25x^2 + y^2 = 25.

Example 16

easy
For y2=xy^2 = x, find dydx\frac{dy}{dx} implicitly.

Example 17

easy
Find ddx[xy]\frac{d}{dx}[xy].

Example 18

easy
Differentiate x2+y2=25x^2 + y^2 = 25 implicitly to find dydx\frac{dy}{dx}.

Example 19

easy
Differentiate x2+y=7x^2 + y = 7 for dydx\frac{dy}{dx}.

Example 20

challenge
Find the tangent line to x2/3+y2/3=4x^{2/3} + y^{2/3} = 4 (astroid) at (1,33)(1, 3\sqrt{3}).
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