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Calculus I · Differentiation Techniques

Implicit Differentiation — Practice Problems and Worked Solutions

When y will not isolate nicely, differentiate the relationship.

The formulas you need here

Implicit differentiation

\frac{d}{d x} [y^{n}] = n y^{n - 1} \frac{d y}{d x}

What it says: Differentiate both sides in x, and every y term hands you a dy/dx by the chain rule.
When to reach for it: Equations you cannot or would rather not rearrange into y = something.
Watch out: Forgetting the dy/dx on a y term is exactly the chain-rule error wearing a different hat.

x^{2} + y^{2} = 25 \Rightarrow \frac{d y}{d x} = - \frac{x}{y}

Collect every dy/dx on one side, then divide.

16 worked problems

Every step is generated and checked by a computer algebra system.

1.
Find dy/dx by implicit differentiation.
$x^{2} + y^{2} - 25 = 0$
Answer: $- \frac{x}{y}$
Step-by-step solution
1. $Differentiate both sides, treating y as a function of x .$
2. $F_{x} + F_{y} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = - \frac{x}{y}$
2.
Find dy/dx by implicit differentiation.
$x y - 6 = 0$
Answer: $- \frac{y}{x}$
Step-by-step solution
1. $Differentiate both sides, treating y as a function of x .$
2. $F_{x} + F_{y} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = - \frac{y}{x}$
3.
Find dy/dx by implicit differentiation.
$x^{2} + x y + y^{2} - 7 = 0$
Answer: $\frac{- 2 x - y}{x + 2 y}$
Step-by-step solution
1. $Differentiate both sides, treating y as a function of x .$
2. $F_{x} + F_{y} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = \frac{- 2 x - y}{x + 2 y}$
4.
Find dy/dx by implicit differentiation.
$x^{3} + y^{3} - 9 = 0$
Answer: $- \frac{x ^{2}}{y ^{2}}$
Step-by-step solution
1. $Differentiate both sides, treating y as a function of x .$
2. $F_{x} + F_{y} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = - \frac{x ^{2}}{y ^{2}}$
5.
Find dy/dx by implicit differentiation.
$y^{2} + sin (x) - 1 = 0$
Answer: $- \frac{c o s ( x )}{2 y}$
Step-by-step solution
1. $Differentiate both sides, treating y as a function of x .$
2. $F_{x} + F_{y} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = - \frac{cos ( x )}{2 y}$
6.
Find dy/dx by implicit differentiation.
$x y + e^{y} - 4 = 0$
Answer: $- \frac{y}{x + e ^{y}}$
Step-by-step solution
1. $Differentiate both sides, treating y as a function of x .$
2. $F_{x} + F_{y} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = - \frac{y}{x + e ^{y}}$
7.
Find dy/dx by implicit differentiation.
$x^{2} + y^{2} - 16 = 0$
Answer: $- \frac{x}{y}$
Step-by-step solution
1. $Differentiate both sides, treating y as a function of x .$
2. $F_{x} + F_{y} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = - \frac{x}{y}$
8.
Find dy/dx by implicit differentiation.
$x y - 12 = 0$
Answer: $- \frac{y}{x}$
Step-by-step solution
1. $Differentiate both sides, treating y as a function of x .$
2. $F_{x} + F_{y} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = - \frac{y}{x}$
9.
Find dy/dx by implicit differentiation.
$x^{2} - y^{2} - 5 = 0$
Answer: $\frac{x}{y}$
Step-by-step solution
1. $Differentiate both sides, treating y as a function of x .$
2. $F_{x} + F_{y} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = \frac{x}{y}$
10.
Find dy/dx by implicit differentiation.
$x^{2} + 3 y^{2} - 12 = 0$
Answer: $- \frac{x}{3 y}$
Step-by-step solution
1. $Differentiate both sides, treating y as a function of x .$
2. $F_{x} + F_{y} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = - \frac{x}{3 y}$
11.
Find dy/dx by implicit differentiation.
$x^{3} + y^{2} - 8 = 0$
Answer: $- \frac{3 x ^{2}}{2 y}$
Step-by-step solution
1. $Differentiate both sides, treating y as a function of x .$
2. $F_{x} + F_{y} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = - \frac{3 x ^{2}}{2 y}$
12.
Find dy/dx by implicit differentiation.
$x y^{2} - 9 = 0$
Answer: $- \frac{y}{2 x}$
Step-by-step solution
1. $Differentiate both sides, treating y as a function of x .$
2. $F_{x} + F_{y} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = - \frac{y}{2 x}$
13.
Find dy/dx by implicit differentiation.
$y^{3} + cos (x) - 2 = 0$
Answer: $\frac{s i n ( x )}{3 y ^{2}}$
Step-by-step solution
1. $Differentiate both sides, treating y as a function of x .$
2. $F_{x} + F_{y} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = \frac{sin ( x )}{3 y ^{2}}$
14.
Find dy/dx by implicit differentiation.
$x^{2} y - 6 = 0$
Answer: $- \frac{2 y}{x}$
Step-by-step solution
1. $Differentiate both sides, treating y as a function of x .$
2. $F_{x} + F_{y} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = - \frac{2 y}{x}$
15.
Find dy/dx by implicit differentiation.
$2 x^{2} + x y - y^{2} - 3 = 0$
Answer: $\frac{- 4 x - y}{x - 2 y}$
Step-by-step solution
1. $Differentiate both sides, treating y as a function of x .$
2. $F_{x} + F_{y} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = \frac{- 4 x - y}{x - 2 y}$
16.
Find dy/dx by implicit differentiation.
$x^{2} + sin (y) - 1 = 0$
Answer: $- \frac{2 x}{c o s ( y )}$
Step-by-step solution
1. $Differentiate both sides, treating y as a function of x .$
2. $F_{x} + F_{y} \frac{d y}{d x} = 0 \Rightarrow \frac{d y}{d x} = - \frac{2 x}{cos ( y )}$

The formulas you need here

Implicit differentiation

16 worked problems

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