Calculus Examples

$\frac{d}{d x} ((x^{2} + 4) (2 x^{3} + 7 x))$

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 $f (x) = x^{2} + 4$ and $g (x) = 2 x^{3} + 7 x$ .

$(x^{2} + 4) \frac{d}{d x} [2 x^{3} + 7 x] + (2 x^{3} + 7 x) \frac{d}{d x} [x^{2} + 4]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $2 x^{3} + 7 x$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [7 x]$ .

$(x^{2} + 4) (\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [7 x]) + (2 x^{3} + 7 x) \frac{d}{d x} [x^{2} + 4]$

Step 2.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

$(x^{2} + 4) (2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [7 x]) + (2 x^{3} + 7 x) \frac{d}{d x} [x^{2} + 4]$

Step 2.3

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

$(x^{2} + 4) (2 (3 x^{2}) + \frac{d}{d x} [7 x]) + (2 x^{3} + 7 x) \frac{d}{d x} [x^{2} + 4]$

Step 2.4

Multiply $3$ by $2$ .

$(x^{2} + 4) (6 x^{2} + \frac{d}{d x} [7 x]) + (2 x^{3} + 7 x) \frac{d}{d x} [x^{2} + 4]$

Step 2.5

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$(x^{2} + 4) (6 x^{2} + 7 \frac{d}{d x} [x]) + (2 x^{3} + 7 x) \frac{d}{d x} [x^{2} + 4]$

Step 2.6

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

$(x^{2} + 4) (6 x^{2} + 7 \cdot 1) + (2 x^{3} + 7 x) \frac{d}{d x} [x^{2} + 4]$

Step 2.7

Multiply $7$ by $1$ .

$(x^{2} + 4) (6 x^{2} + 7) + (2 x^{3} + 7 x) \frac{d}{d x} [x^{2} + 4]$

Step 2.8

By the Sum Rule, the derivative of $x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$ .

$(x^{2} + 4) (6 x^{2} + 7) + (2 x^{3} + 7 x) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4])$

Step 2.9

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

$(x^{2} + 4) (6 x^{2} + 7) + (2 x^{3} + 7 x) (2 x + \frac{d}{d x} [4])$

Step 2.10

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$(x^{2} + 4) (6 x^{2} + 7) + (2 x^{3} + 7 x) (2 x + 0)$

Step 2.11

Simplify the expression.

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Step 2.11.1

Add $2 x$ and $0$ .

$(x^{2} + 4) (6 x^{2} + 7) + (2 x^{3} + 7 x) (2 x)$

Step 2.11.2

Move $2$ to the left of $2 x^{3} + 7 x$ .

$(x^{2} + 4) (6 x^{2} + 7) + 2 \cdot (2 x^{3} + 7 x) x$

Step 3

Simplify.

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Step 3.1