Calculus Examples

$y = (x + 1) (x^{2} - 1)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ((x + 1) (x^{2} - 1))$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x + 1$ and $g (x) = x^{2} - 1$ .

$(x + 1) \frac{d}{d x} [x^{2} - 1] + (x^{2} - 1) \frac{d}{d x} [x + 1]$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

$(x + 1) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]) + (x^{2} - 1) \frac{d}{d x} [x + 1]$

Step 3.2.2

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

$(x + 1) (2 x + \frac{d}{d x} [- 1]) + (x^{2} - 1) \frac{d}{d x} [x + 1]$

Step 3.2.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$(x + 1) (2 x + 0) + (x^{2} - 1) \frac{d}{d x} [x + 1]$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $2 x$ and $0$ .

$(x + 1) (2 x) + (x^{2} - 1) \frac{d}{d x} [x + 1]$

Step 3.2.4.2

Move $2$ to the left of $x + 1$ .

$2 \cdot (x + 1) x + (x^{2} - 1) \frac{d}{d x} [x + 1]$

Step 3.2.5

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$2 (x + 1) x + (x^{2} - 1) (\frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 3.2.6

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

$2 (x + 1) x + (x^{2} - 1) (1 + \frac{d}{d x} [1])$

Step 3.2.7

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 (x + 1) x + (x^{2} - 1) (1 + 0)$

Step 3.2.8

Simplify the expression.

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Step 3.2.8.1

Add $1$ and $0$ .

$2 (x + 1) x + (x^{2} - 1) \cdot 1$

Step 3.2.8.2

Multiply $x^{2} - 1$ by $1$ .

$2 (x + 1) x + x^{2} - 1$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$(2 x + 2 \cdot 1) x + x^{2} - 1$

Step 3.3.2

Apply the distributive property.

$2 x \cdot x + 2 \cdot 1 x + x^{2} - 1$

Step 3.3.3

Combine terms.

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Step 3.3.3.1

Raise $x$ to the power of $1$ .

$2 (x^{1} x) + 2 \cdot 1 x + x^{2} - 1$

Step 3.3.3.2

Raise $x$ to the power of $1$ .

$2 (x^{1} x^{1}) + 2 \cdot 1 x + x^{2} - 1$

Step 3.3.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{1 + 1} + 2 \cdot 1 x + x^{2} - 1$

Step 3.3.3.4

Add $1$ and $1$ .

$2 x^{2} + 2 \cdot 1 x + x^{2} - 1$

Step 3.3.3.5

Multiply $2$ by $1$ .

$2 x^{2} + 2 x + x^{2} - 1$

Step 3.3.3.6

Add $2 x^{2}$ and $x^{2}$ .

$3 x^{2} + 2 x - 1$

Step 4

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

$y' = 3 x^{2} + 2 x - 1$

Step 5

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

$\frac{d y}{d x} = 3 x^{2} + 2 x - 1$