Calculus Examples

$y = {(a x + b)}^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d b} (y) = \frac{d}{d b} ({(a x + b)}^{2})$

Step 2

The derivative of $y$ with respect to $b$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Rewrite ${(a x + b)}^{2}$ as $(a x + b) (a x + b)$ .

$\frac{d}{d b} [(a x + b) (a x + b)]$

Step 3.2

Expand $(a x + b) (a x + b)$ using the FOIL Method.

Tap for more steps...

Step 3.2.1

Apply the distributive property.

$\frac{d}{d b} [a x (a x + b) + b (a x + b)]$

Step 3.2.2

Apply the distributive property.

$\frac{d}{d b} [a x (a x) + a x b + b (a x + b)]$

Step 3.2.3

Apply the distributive property.

$\frac{d}{d b} [a x (a x) + a x b + b (a x) + b \cdot b]$

Step 3.3

Simplify and combine like terms.

Tap for more steps...

Step 3.3.1

Simplify each term.

Tap for more steps...

Step 3.3.1.1

Multiply $a$ by $a$ by adding the exponents.

Tap for more steps...

Step 3.3.1.1.1

Move $a$ .

$\frac{d}{d b} [a \cdot a x \cdot x + a x b + b (a x) + b \cdot b]$

Step 3.3.1.1.2

Multiply $a$ by $a$ .

$\frac{d}{d b} [a^{2} x \cdot x + a x b + b (a x) + b \cdot b]$

Step 3.3.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.3.1.2.1

Move $x$ .

$\frac{d}{d b} [a^{2} (x \cdot x) + a x b + b (a x) + b \cdot b]$

Step 3.3.1.2.2

Multiply $x$ by $x$ .

$\frac{d}{d b} [a^{2} x^{2} + a x b + b (a x) + b \cdot b]$

Step 3.3.1.3

Multiply $b$ by $b$ .

$\frac{d}{d b} [a^{2} x^{2} + a x b + b a x + b^{2}]$

Step 3.3.2

Reorder $b$ and $a$ .

$\frac{d}{d b} [a^{2} x^{2} + a b x + a b x + b^{2}]$

Step 3.3.3

Add $a b x$ and $a b x$ .

$\frac{d}{d b} [a^{2} x^{2} + 2 a b x + b^{2}]$

Step 3.4

By the Sum Rule, the derivative of $a^{2} x^{2} + 2 a b x + b^{2}$ with respect to $b$ is $\frac{d}{d b} [a^{2} x^{2}] + \frac{d}{d b} [2 a b x] + \frac{d}{d b} [b^{2}]$ .

$\frac{d}{d b} [a^{2} x^{2}] + \frac{d}{d b} [2 a b x] + \frac{d}{d b} [b^{2}]$

Step 3.5

Since $a^{2} x^{2}$ is constant with respect to $b$ , the derivative of $a^{2} x^{2}$ with respect to $b$ is $0$ .

$0 + \frac{d}{d b} [2 a b x] + \frac{d}{d b} [b^{2}]$

Step 3.6

Add $0$ and $\frac{d}{d b} [2 a b x]$ .

$\frac{d}{d b} [2 a b x] + \frac{d}{d b} [b^{2}]$

Step 3.7

Since $2 a x$ is constant with respect to $b$ , the derivative of $2 a b x$ with respect to $b$ is $2 a x \frac{d}{d b} [b]$ .

$2 a x \frac{d}{d b} [b] + \frac{d}{d b} [b^{2}]$

Step 3.8

Differentiate using the Power Rule which states that $\frac{d}{d b} [b^{n}]$ is $n b^{n - 1}$ where $n = 1$ .

$2 a x \cdot 1 + \frac{d}{d b} [b^{2}]$

Step 3.9

Multiply $2$ by $1$ .

$2 a x + \frac{d}{d b} [b^{2}]$

Step 3.10

Differentiate using the Power Rule which states that $\frac{d}{d b} [b^{n}]$ is $n b^{n - 1}$ where $n = 2$ .

$2 a x + 2 b$

Step 3.11

Reorder terms.

$2 b + 2 a x$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 2 b + 2 a x$

Step 5

Replace $y'$ with $\frac{d y}{d b}$ .

$\frac{d y}{d b} = 2 b + 2 a x$