dy/dx = -(3x^2 - 2y)/(3y^2 - 2x)

Implicit differentiation worksheet

Differentiate both sides as usual, but every time you differentiate a y you pick up a dy/dx from the chain rule. Then collect those terms and solve for it.

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Pick more than one for a sheet that mixes them.

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Sigma Prep · sigmaprep.io/worksheets Name Implicit Differentiation Date Period Find each derivative, or answer as asked. Find $\frac{dy}{dx}$ $e^{y} + 4 y = x^{3} + 2 x - 32$ $x^{2} y + y^{3} = 5$ $3 x^{2} - x y + 2 y^{2} = 32$ $\ln y + 4 y = - x^{2} + 13$ $x y^{2} - 3 y = 14$ $x^{3} + y^{3} = 2 x y + 2$ $x^{2} y^{2} + 4 x = 8$ Find $\frac{dy}{dx}$ at the given point Find $\frac{d y}{d x}$ at $(- 1, - 2)$ on $x^{2} y - y^{2} = - 6$ Find $\frac{d y}{d x}$ at $(3, - 3)$ on $x y - 2 y = 3 x^{2} - 30$ Find $\frac{d y}{d x}$ at $(- 1, - 1)$ on $x^{3} + y^{3} = 3 x y - 5$ Find $\frac{d y}{d x}$ at $(1, 0)$ on $\cos y - y = - 4 x + 5$ Find $\frac{d y}{d x}$ at $(2, 2)$ on $x^{2} y^{2} + 5 x = 26$ Find $\frac{d y}{d x}$ at $(1, 1)$ on $\ln y + 2 y = x^{2} + 3 x - 2$ Find $\frac{d y}{d x}$ at $(3, - 1)$ on $x y^{2} + x^{3} = 30$ Find the points where the tangent line is horizontal Find the points on $x^{2} y + 4 y^{3} = - 32$ where the tangent line is horizontal. Find the points on $x^{2} + 2 x y + 4 y^{2} = 48$ where the tangent line is horizontal. Find the points on $2 x y^{2} - 2 x^{2} = 8$ where the tangent line is horizontal. Find the points on $x^{2} + y^{2} - 4 x + 2 y = 4$ where the tangent line is horizontal. Find the points on $x^{2} y + 4 x = - 2$ where the tangent line is horizontal. Find the points on $y^{2} = x^{3} - 9 x^{2} + 15 x + 2$ where the tangent line is horizontal. Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Implicit Differentiation

Date Period

Find each derivative, or answer as asked.

Find $\frac{dy}{dx}$
$e^{y} + 4 y = x^{3} + 2 x - 32$
$x^{2} y + y^{3} = 5$
$3 x^{2} - x y + 2 y^{2} = 32$
$\ln y + 4 y = - x^{2} + 13$
$x y^{2} - 3 y = 14$
$x^{3} + y^{3} = 2 x y + 2$
$x^{2} y^{2} + 4 x = 8$

Find $\frac{dy}{dx}$ at the given point
Find $\frac{d y}{d x}$ at $(- 1, - 2)$ on $x^{2} y - y^{2} = - 6$
Find $\frac{d y}{d x}$ at $(3, - 3)$ on $x y - 2 y = 3 x^{2} - 30$
Find $\frac{d y}{d x}$ at $(- 1, - 1)$ on $x^{3} + y^{3} = 3 x y - 5$
Find $\frac{d y}{d x}$ at $(1, 0)$ on $\cos y - y = - 4 x + 5$
Find $\frac{d y}{d x}$ at $(2, 2)$ on $x^{2} y^{2} + 5 x = 26$
Find $\frac{d y}{d x}$ at $(1, 1)$ on $\ln y + 2 y = x^{2} + 3 x - 2$
Find $\frac{d y}{d x}$ at $(3, - 1)$ on $x y^{2} + x^{3} = 30$

Find the points where the tangent line is horizontal
Find the points on $x^{2} y + 4 y^{3} = - 32$ where the tangent line is horizontal.
Find the points on $x^{2} + 2 x y + 4 y^{2} = 48$ where the tangent line is horizontal.
Find the points on $2 x y^{2} - 2 x^{2} = 8$ where the tangent line is horizontal.
Find the points on $x^{2} + y^{2} - 4 x + 2 y = 4$ where the tangent line is horizontal.
Find the points on $x^{2} y + 4 x = - 2$ where the tangent line is horizontal.
Find the points on $y^{2} = x^{3} - 9 x^{2} + 15 x + 2$ where the tangent line is horizontal.



Sigma Prep · sigmaprep.io/worksheets Name Implicit DifferentiationAnswer keyVersion 1 Date Period Find each derivative, or answer as asked. Find $\frac{dy}{dx}$ $e^{y} + 4 y = x^{3} + 2 x - 32$ $\frac{d y}{d x} = \frac{3 x^{2} + 2}{e^{y} + 4}$ $x^{2} y + y^{3} = 5$ $\frac{d y}{d x} = - \frac{2 x y}{x^{2} + 3 y^{2}}$ $3 x^{2} - x y + 2 y^{2} = 32$ $\frac{d y}{d x} = \frac{6 x - y}{x - 4 y}$ $\ln y + 4 y = - x^{2} + 13$ $\frac{d y}{d x} = - \frac{2 x y}{4 y + 1}$ $x y^{2} - 3 y = 14$ $\frac{d y}{d x} = - \frac{y^{2}}{2 x y - 3}$ $x^{3} + y^{3} = 2 x y + 2$ $\frac{d y}{d x} = - \frac{3 x^{2} - 2 y}{3 y^{2} - 2 x}$ $x^{2} y^{2} + 4 x = 8$ $\frac{d y}{d x} = - \frac{x y^{2} + 2}{x^{2} y}$ Find $\frac{dy}{dx}$ at the given point Find $\frac{d y}{d x}$ at $(- 1, - 2)$ on $x^{2} y - y^{2} = - 6$ $- \frac{4}{5}$ Find $\frac{d y}{d x}$ at $(3, - 3)$ on $x y - 2 y = 3 x^{2} - 30$ $21$ Find $\frac{d y}{d x}$ at $(- 1, - 1)$ on $x^{3} + y^{3} = 3 x y - 5$ $- 1$ Find $\frac{d y}{d x}$ at $(1, 0)$ on $\cos y - y = - 4 x + 5$ $4$ Find $\frac{d y}{d x}$ at $(2, 2)$ on $x^{2} y^{2} + 5 x = 26$ $- \frac{21}{16}$ Find $\frac{d y}{d x}$ at $(1, 1)$ on $\ln y + 2 y = x^{2} + 3 x - 2$ $\frac{5}{3}$ Find $\frac{d y}{d x}$ at $(3, - 1)$ on $x y^{2} + x^{3} = 30$ $\frac{14}{3}$ Find the points where the tangent line is horizontal Find the points on $x^{2} y + 4 y^{3} = - 32$ where the tangent line is horizontal. $(0, - 2)$ Find the points on $x^{2} + 2 x y + 4 y^{2} = 48$ where the tangent line is horizontal. $(- 4, 4), (4, - 4)$ Find the points on $2 x y^{2} - 2 x^{2} = 8$ where the tangent line is horizontal. $(2, - 2), (2, 2)$ Find the points on $x^{2} + y^{2} - 4 x + 2 y = 4$ where the tangent line is horizontal. $(2, - 4), (2, 2)$ Find the points on $x^{2} y + 4 x = - 2$ where the tangent line is horizontal. $(- 1, 2)$ Find the points on $y^{2} = x^{3} - 9 x^{2} + 15 x + 2$ where the tangent line is horizontal. $(1, - 3), (1, 3)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Implicit DifferentiationAnswer keyVersion 1

Date Period

Find each derivative, or answer as asked.

Find $\frac{dy}{dx}$
$e^{y} + 4 y = x^{3} + 2 x - 32$ $\frac{d y}{d x} = \frac{3 x^{2} + 2}{e^{y} + 4}$
$x^{2} y + y^{3} = 5$ $\frac{d y}{d x} = - \frac{2 x y}{x^{2} + 3 y^{2}}$
$3 x^{2} - x y + 2 y^{2} = 32$ $\frac{d y}{d x} = \frac{6 x - y}{x - 4 y}$
$\ln y + 4 y = - x^{2} + 13$ $\frac{d y}{d x} = - \frac{2 x y}{4 y + 1}$
$x y^{2} - 3 y = 14$ $\frac{d y}{d x} = - \frac{y^{2}}{2 x y - 3}$
$x^{3} + y^{3} = 2 x y + 2$ $\frac{d y}{d x} = - \frac{3 x^{2} - 2 y}{3 y^{2} - 2 x}$
$x^{2} y^{2} + 4 x = 8$ $\frac{d y}{d x} = - \frac{x y^{2} + 2}{x^{2} y}$

Find $\frac{dy}{dx}$ at the given point
Find $\frac{d y}{d x}$ at $(- 1, - 2)$ on $x^{2} y - y^{2} = - 6$ $- \frac{4}{5}$
Find $\frac{d y}{d x}$ at $(3, - 3)$ on $x y - 2 y = 3 x^{2} - 30$ $21$
Find $\frac{d y}{d x}$ at $(- 1, - 1)$ on $x^{3} + y^{3} = 3 x y - 5$ $- 1$
Find $\frac{d y}{d x}$ at $(1, 0)$ on $\cos y - y = - 4 x + 5$ $4$
Find $\frac{d y}{d x}$ at $(2, 2)$ on $x^{2} y^{2} + 5 x = 26$ $- \frac{21}{16}$
Find $\frac{d y}{d x}$ at $(1, 1)$ on $\ln y + 2 y = x^{2} + 3 x - 2$ $\frac{5}{3}$
Find $\frac{d y}{d x}$ at $(3, - 1)$ on $x y^{2} + x^{3} = 30$ $\frac{14}{3}$

Find the points where the tangent line is horizontal
Find the points on $x^{2} y + 4 y^{3} = - 32$ where the tangent line is horizontal. $(0, - 2)$
Find the points on $x^{2} + 2 x y + 4 y^{2} = 48$ where the tangent line is horizontal. $(- 4, 4), (4, - 4)$
Find the points on $2 x y^{2} - 2 x^{2} = 8$ where the tangent line is horizontal. $(2, - 2), (2, 2)$
Find the points on $x^{2} + y^{2} - 4 x + 2 y = 4$ where the tangent line is horizontal. $(2, - 4), (2, 2)$
Find the points on $x^{2} y + 4 x = - 2$ where the tangent line is horizontal. $(- 1, 2)$
Find the points on $y^{2} = x^{3} - 9 x^{2} + 15 x + 2$ where the tangent line is horizontal. $(1, - 3), (1, 3)$

How to do these

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

Differentiating gives 2x plus 2y dy/dx.
The right side is a constant, so it gives zero.
Solve for dy/dx.

The answer is $\frac{d y}{d x} = - \frac{x}{y}$ .

Where students go wrong

Differentiating y as though it were a number. It is a function of x, so it brings dy/dx with it every single time, and missing one leaves an equation you cannot solve.

Also called dy/dx, differentiating implicitly or implicit derivatives.

What you can put on this worksheet

Find dy/dx: separate terms (easy), products and e^y, ln y or trig of y (medium), functions of xy and x + y (hard): $x^{2} + y^{2} = 25$ → $\frac{d y}{d x} = - \frac{x}{y}$
The derivative at a point: dy/dx (easy, medium), d²y/dx² (hard): Find $\frac{d y}{d x}$ at $(3, 4)$ on $x^{2} + y^{2} = 25$ → $- \frac{3}{4}$
Tangents: the slope where x is given (easy), horizontal tangents (medium), tangent lines and horizontal and vertical tangents (hard): Find $\frac{d y}{d x}$ on ${(x - 3)}^{2} + 3 {(y + 2)}^{2} = 13$ where $x = 4$ and $y > - 2$ → $- \frac{1}{6}$

Questions about these worksheets

Yes. Every implicit differentiation sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the implicit differentiation questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 3: find dy/dx: separate terms (easy), products and e^y, ln y or trig of y (medium), functions of xy and x + y (hard), the derivative at a point: dy/dx (easy, medium), d²y/dx² (hard) and tangents: the slope where x is given (easy), horizontal tangents (medium), tangent lines and horizontal and vertical tangents (hard). Tick as many as you want and set how many of each, or let it spread them evenly.

Differentiating y as though it were a number. It is a function of x, so it brings dy/dx with it every single time, and missing one leaves an equation you cannot solve.

Implicit differentiation worksheet

Select your difficulty

Find dydx\frac{dy}{dx}dxdy​

Find dydx\frac{dy}{dx}dxdy​ at the given point

Find the points where the tangent line is horizontal

Find dydx\frac{dy}{dx}dxdy​

Find dydx\frac{dy}{dx}dxdy​ at the given point

Find the points where the tangent line is horizontal

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these implicit differentiation worksheets free?+

Do the implicit differentiation worksheets come with an answer key?+

Are these implicit differentiation worksheets printable?+

What grade are these implicit differentiation worksheets for?+

Can I get a different version of the same implicit differentiation worksheet?+

Is this a free alternative to Kuta Software for implicit differentiation?+

Can I choose how hard the implicit differentiation questions are?+

What kinds of question are on a implicit differentiation worksheet?+

What do students get wrong with implicit differentiation?+

Find $\frac{dy}{dx}$

Find $\frac{dy}{dx}$ at the given point

Find $\frac{dy}{dx}$

Find $\frac{dy}{dx}$ at the given point