Calculus Examples

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

Step 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 $f (x) = x^{2} - 1$ and $g (x) = x^{3} + 3$ .

$(x^{2} - 1) \frac{d}{d x} [x^{3} + 3] + (x^{3} + 3) \frac{d}{d x} [x^{2} - 1]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{3} + 3$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3]$ .

$(x^{2} - 1) (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3]) + (x^{3} + 3) \frac{d}{d x} [x^{2} - 1]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$(x^{2} - 1) (3 x^{2} + \frac{d}{d x} [3]) + (x^{3} + 3) \frac{d}{d x} [x^{2} - 1]$

Step 2.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$(x^{2} - 1) (3 x^{2} + 0) + (x^{3} + 3) \frac{d}{d x} [x^{2} - 1]$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $3 x^{2}$ and $0$ .

$(x^{2} - 1) (3 x^{2}) + (x^{3} + 3) \frac{d}{d x} [x^{2} - 1]$

Step 2.4.2

Move $3$ to the left of $x^{2} - 1$ .

$3 \cdot (x^{2} - 1) x^{2} + (x^{3} + 3) \frac{d}{d x} [x^{2} - 1]$

Step 2.5

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

$3 (x^{2} - 1) x^{2} + (x^{3} + 3) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1])$

Step 2.6

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

$3 (x^{2} - 1) x^{2} + (x^{3} + 3) (2 x + \frac{d}{d x} [- 1])$

Step 2.7

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

$3 (x^{2} - 1) x^{2} + (x^{3} + 3) (2 x + 0)$

Step 2.8

Simplify the expression.

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Step 2.8.1

Add $2 x$ and $0$ .

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

Step 2.8.2

Move $2$ to the left of $x^{3} + 3$ .

$3 (x^{2} - 1) x^{2} + 2 \cdot (x^{3} + 3) x$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$(3 x^{2} + 3 \cdot - 1) x^{2} + 2 (x^{3} + 3) x$

Step 3.2

Apply the distributive property.

$3 x^{2} x^{2} + 3 \cdot - 1 x^{2} + 2 (x^{3} + 3) x$

Step 3.3

Apply the distributive property.

$3 x^{2} x^{2} + 3 \cdot - 1 x^{2} + (2 x^{3} + 2 \cdot 3) x$

Step 3.4

Apply the distributive property.

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

Step 3.5

Combine terms.

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Step 3.5.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.5.1.1

Move $x^{2}$ .

$3 (x^{2} x^{2}) + 3 \cdot - 1 x^{2} + 2 x^{3} x + 2 \cdot 3 x$

Step 3.5.1.2

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

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

Step 3.5.1.3

Add $2$ and $2$ .

$3 x^{4} + 3 \cdot - 1 x^{2} + 2 x^{3} x + 2 \cdot 3 x$

Step 3.5.2

Multiply $3$ by $- 1$ .

$3 x^{4} - 3 x^{2} + 2 x^{3} x + 2 \cdot 3 x$

Step 3.5.3

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

$3 x^{4} - 3 x^{2} + 2 (x^{1} x^{3}) + 2 \cdot 3 x$

Step 3.5.4

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

$3 x^{4} - 3 x^{2} + 2 x^{1 + 3} + 2 \cdot 3 x$

Step 3.5.5

Add $1$ and $3$ .

$3 x^{4} - 3 x^{2} + 2 x^{4} + 2 \cdot 3 x$

Step 3.5.6

Multiply $2$ by $3$ .

$3 x^{4} - 3 x^{2} + 2 x^{4} + 6 x$

Step 3.5.7

Add $3 x^{4}$ and $2 x^{4}$ .

$5 x^{4} - 3 x^{2} + 6 x$