Calculus · Step-by-step guide

Implicit Differentiation

Differentiate equations where y is not isolated, including tangent slopes and second derivatives.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

Differentiate both sides with respect to x. Every derivative of a y-expression needs a factor of dy/dx\displaystyle dy/dx by the chain rule. Collect those terms and solve algebraically for dy/dx\displaystyle dy/dx.

01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Treat y as a function

Because y depends on x, d(yn)/dx=nyn−1y′\displaystyle d(y^n)/dx=ny^{n-1}y'.

2

Differentiate every term

Apply product and chain rules wherever x and y appear together.

3

Solve for y′

Move all terms containing y′ to one side, factor y′, and divide.

02

See the structure

A picture makes the relationships easier to remember.

tangent line
The derivative at the marked point is the slope of the highlighted tangent line.
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Circle

For x2+y2=25\displaystyle x^2+y^2=25, find y′.

  1. Differentiate: 2x+2yy′=0\displaystyle 2x+2yy'=0.
  2. Move 2x: 2yy′=−2x\displaystyle 2yy'=-2x.
  3. Divide by 2y.
Answery′=−x/y\displaystyle y'=-x/y
Example 2Mixed product

Differentiate xy+y2=10\displaystyle xy+y^2=10.

  1. Product rule gives xy′+y\displaystyle xy'+y.
  2. Derivative of y² is 2yy′\displaystyle 2yy'.
  3. Collect: y′(x+2y)=−y\displaystyle y'(x+2y)=-y.
Answery′=−y/(x+2y)\displaystyle y'=-y/(x+2y)
Example 3Slope at a point

Find slope on x2+y2=25\displaystyle x^2+y^2=25 at (3,4)\displaystyle (3,4).

  1. Use y′=−x/y\displaystyle y'=-x/y.
  2. Substitute x = 3, y = 4.
  3. Keep the exact ratio.
Answer−3/4\displaystyle -3/4
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Writing the derivative of y² as 2y without y′.

  • Using the ordinary product rule incorrectly on xy.

  • Substituting coordinates before solving for y′ when it complicates algebra.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Implicit Differentiation

Practice chain-rule factors, tangent slopes, and implicit curves.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

For x2+y2=25\displaystyle x^2+y^2=25, find dy/dx\displaystyle dy/dx.

Show hint

Differentiate y² with the chain rule.

Show worked solution
  1. 2x+2y(dy/dx)=0\displaystyle 2x+2y(dy/dx)=0.
  2. 2y(dy/dx)=−2x\displaystyle 2y(dy/dx)=-2x.

Answer: dy/dx=−x/y\displaystyle dy/dx=-x/y

2

Find the slope on x2+y2=25\displaystyle x^2+y^2=25 at (3,4)\displaystyle (3,4).

Show hint

Use dy/dx=−x/y\displaystyle dy/dx=-x/y.

Show worked solution
  1. Substitute x=3,y=4\displaystyle x=3,y=4.
  2. dy/dx=−3/4\displaystyle dy/dx=-3/4.

Answer: −3/4\displaystyle -3/4

3

Differentiate xy=6\displaystyle xy=6.

Show hint

Use the product rule on xy.

Show worked solution
  1. x(dy/dx)+y=0\displaystyle x(dy/dx)+y=0.
  2. Solve for dy/dx\displaystyle dy/dx.

Answer: dy/dx=−y/x\displaystyle dy/dx=-y/x

4

Differentiate x3+y3=9\displaystyle x^3+y^3=9.

Show hint

Each y derivative gets y′.

Show worked solution
  1. 3x2+3y2y′=0\displaystyle 3x^2+3y^2y'=0.
  2. 3y2y′=−3x2\displaystyle 3y^2y'=-3x^2.

Answer: y′=−x2/y2\displaystyle y'=-x^2/y^2

5

Differentiate x2+xy+y2=7\displaystyle x^2+xy+y^2=7.

Show hint

Use product rule on xy.

Show worked solution
  1. 2x+(xy′+y)+2yy′=0\displaystyle 2x+(xy'+y)+2yy'=0.
  2. Group: (x+2y)y′=−(2x+y)\displaystyle (x+2y)y'=-(2x+y).

Answer: y′=−(2x+y)/(x+2y)\displaystyle y'=-(2x+y)/(x+2y)

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

Differentiate both sides with respect to x. Every derivative of a y-expression needs a factor of dy/dx\displaystyle dy/dx by the chain rule. Collect those terms and solve algebraically for dy/dx\displaystyle dy/dx.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

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