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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Sigma Prep · sigmaprep.io/worksheets Name Implicit Differentiation Date Period Find dy/dx for each equation. $4 x^{3} + 5 y^{2} - 5 x y = 10$ $- 4 x^{2} - 4 y^{2} + 2 x y = 4$ $- 4 x^{2} + 5 y^{2} + x y = 15$ $4 x^{2} - 4 y^{3} - x y = 15$ $5 x^{3} - 5 y^{2} + 3 x y = 7$ $5 x^{3} + 4 y^{3} - 5 x y = 6$ $- 5 x^{3} + 4 y^{3} + x y = 8$ $5 x^{2} - 5 y^{2} + 2 x y = 4$ $5 x^{2} - 4 y^{3} - x y = 7$ $4 x^{3} + 4 y^{3} + 5 x y = 18$ $- 5 x^{2} + 4 y^{2} - 5 x y = 16$ $- 5 x^{3} - 5 y^{2} - 5 x y = 18$ $- 4 x^{3} - 5 y^{3} + x y = 7$ $- 4 x^{2} + 4 y^{2} + 3 x y = 9$ $5 x^{3} - 4 y^{2} + 4 x y = 1$ $- 4 x^{2} + 4 y^{2} + 3 x y = 12$ $- 5 x^{3} - 5 y^{2} - 3 x y = 19$ $- 5 x^{2} - 5 y^{2} - x y = 20$ $- 5 x^{3} + 4 y^{2} + x y = 6$ $- 5 x^{3} - 4 y^{3} + 4 x y = 1$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Implicit Differentiation

Date Period

Find dy/dx for each equation.

$4 x^{3} + 5 y^{2} - 5 x y = 10$
$- 4 x^{2} - 4 y^{2} + 2 x y = 4$
$- 4 x^{2} + 5 y^{2} + x y = 15$
$4 x^{2} - 4 y^{3} - x y = 15$
$5 x^{3} - 5 y^{2} + 3 x y = 7$
$5 x^{3} + 4 y^{3} - 5 x y = 6$
$- 5 x^{3} + 4 y^{3} + x y = 8$
$5 x^{2} - 5 y^{2} + 2 x y = 4$
$5 x^{2} - 4 y^{3} - x y = 7$
$4 x^{3} + 4 y^{3} + 5 x y = 18$
$- 5 x^{2} + 4 y^{2} - 5 x y = 16$
$- 5 x^{3} - 5 y^{2} - 5 x y = 18$
$- 4 x^{3} - 5 y^{3} + x y = 7$
$- 4 x^{2} + 4 y^{2} + 3 x y = 9$
$5 x^{3} - 4 y^{2} + 4 x y = 1$
$- 4 x^{2} + 4 y^{2} + 3 x y = 12$
$- 5 x^{3} - 5 y^{2} - 3 x y = 19$
$- 5 x^{2} - 5 y^{2} - x y = 20$
$- 5 x^{3} + 4 y^{2} + x y = 6$
$- 5 x^{3} - 4 y^{3} + 4 x y = 1$



Sigma Prep · sigmaprep.io/worksheets Name Implicit DifferentiationAnswer keyVersion 1 Date Period Find dy/dx for each equation. $4 x^{3} + 5 y^{2} - 5 x y = 10$ $\frac{- 12 x^{2} + 5 y}{10 y - 5 x}$ $- 4 x^{2} - 4 y^{2} + 2 x y = 4$ $\frac{- 4 x + y}{4 y - x}$ $- 4 x^{2} + 5 y^{2} + x y = 15$ $\frac{8 x - y}{10 y + x}$ $4 x^{2} - 4 y^{3} - x y = 15$ $\frac{8 x - y}{12 y^{2} + x}$ $5 x^{3} - 5 y^{2} + 3 x y = 7$ $\frac{15 x^{2} + 3 y}{10 y - 3 x}$ $5 x^{3} + 4 y^{3} - 5 x y = 6$ $\frac{- 15 x^{2} + 5 y}{12 y^{2} - 5 x}$ $- 5 x^{3} + 4 y^{3} + x y = 8$ $\frac{15 x^{2} - y}{12 y^{2} + x}$ $5 x^{2} - 5 y^{2} + 2 x y = 4$ $\frac{5 x + y}{5 y - x}$ $5 x^{2} - 4 y^{3} - x y = 7$ $\frac{10 x - y}{12 y^{2} + x}$ $4 x^{3} + 4 y^{3} + 5 x y = 18$ $\frac{- 12 x^{2} - 5 y}{12 y^{2} + 5 x}$ $- 5 x^{2} + 4 y^{2} - 5 x y = 16$ $\frac{10 x + 5 y}{8 y - 5 x}$ $- 5 x^{3} - 5 y^{2} - 5 x y = 18$ $\frac{- 3 x^{2} - y}{2 y + x}$ $- 4 x^{3} - 5 y^{3} + x y = 7$ $\frac{- 12 x^{2} + y}{15 y^{2} - x}$ $- 4 x^{2} + 4 y^{2} + 3 x y = 9$ $\frac{8 x - 3 y}{8 y + 3 x}$ $5 x^{3} - 4 y^{2} + 4 x y = 1$ $\frac{15 x^{2} + 4 y}{8 y - 4 x}$ $- 4 x^{2} + 4 y^{2} + 3 x y = 12$ $\frac{8 x - 3 y}{8 y + 3 x}$ $- 5 x^{3} - 5 y^{2} - 3 x y = 19$ $\frac{- 15 x^{2} - 3 y}{10 y + 3 x}$ $- 5 x^{2} - 5 y^{2} - x y = 20$ $\frac{- 10 x - y}{10 y + x}$ $- 5 x^{3} + 4 y^{2} + x y = 6$ $\frac{15 x^{2} - y}{8 y + x}$ $- 5 x^{3} - 4 y^{3} + 4 x y = 1$ $\frac{- 15 x^{2} + 4 y}{12 y^{2} - 4 x}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Implicit DifferentiationAnswer keyVersion 1

Date Period

Find dy/dx for each equation.

$4 x^{3} + 5 y^{2} - 5 x y = 10$ $\frac{- 12 x^{2} + 5 y}{10 y - 5 x}$
$- 4 x^{2} - 4 y^{2} + 2 x y = 4$ $\frac{- 4 x + y}{4 y - x}$
$- 4 x^{2} + 5 y^{2} + x y = 15$ $\frac{8 x - y}{10 y + x}$
$4 x^{2} - 4 y^{3} - x y = 15$ $\frac{8 x - y}{12 y^{2} + x}$
$5 x^{3} - 5 y^{2} + 3 x y = 7$ $\frac{15 x^{2} + 3 y}{10 y - 3 x}$
$5 x^{3} + 4 y^{3} - 5 x y = 6$ $\frac{- 15 x^{2} + 5 y}{12 y^{2} - 5 x}$
$- 5 x^{3} + 4 y^{3} + x y = 8$ $\frac{15 x^{2} - y}{12 y^{2} + x}$
$5 x^{2} - 5 y^{2} + 2 x y = 4$ $\frac{5 x + y}{5 y - x}$
$5 x^{2} - 4 y^{3} - x y = 7$ $\frac{10 x - y}{12 y^{2} + x}$
$4 x^{3} + 4 y^{3} + 5 x y = 18$ $\frac{- 12 x^{2} - 5 y}{12 y^{2} + 5 x}$
$- 5 x^{2} + 4 y^{2} - 5 x y = 16$ $\frac{10 x + 5 y}{8 y - 5 x}$
$- 5 x^{3} - 5 y^{2} - 5 x y = 18$ $\frac{- 3 x^{2} - y}{2 y + x}$
$- 4 x^{3} - 5 y^{3} + x y = 7$ $\frac{- 12 x^{2} + y}{15 y^{2} - x}$
$- 4 x^{2} + 4 y^{2} + 3 x y = 9$ $\frac{8 x - 3 y}{8 y + 3 x}$
$5 x^{3} - 4 y^{2} + 4 x y = 1$ $\frac{15 x^{2} + 4 y}{8 y - 4 x}$
$- 4 x^{2} + 4 y^{2} + 3 x y = 12$ $\frac{8 x - 3 y}{8 y + 3 x}$
$- 5 x^{3} - 5 y^{2} - 3 x y = 19$ $\frac{- 15 x^{2} - 3 y}{10 y + 3 x}$
$- 5 x^{2} - 5 y^{2} - x y = 20$ $\frac{- 10 x - y}{10 y + x}$
$- 5 x^{3} + 4 y^{2} + x y = 6$ $\frac{15 x^{2} - y}{8 y + x}$
$- 5 x^{3} - 4 y^{3} + 4 x y = 1$ $\frac{- 15 x^{2} + 4 y}{12 y^{2} - 4 x}$

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 $- \frac{x}{y}$ .

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.

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.

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

Implicit Differentiation

Implicit DifferentiationAnswer keyVersion 1

How to do these

Where students go wrong

Questions about these worksheets

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