Calculus Examples

$f (x) = 5 x^{5} + 7 x^{2} \cdot (2 x^{4}) + 5$

Step 1

By the Sum Rule, the derivative of $5 x^{5} + 7 x^{2} \cdot (2 x^{4}) + 5$ with respect to $x$ is $\frac{d}{d x} [5 x^{5}] + \frac{d}{d x} [7 x^{2} \cdot (2 x^{4})] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [5 x^{5}] + \frac{d}{d x} [7 x^{2} \cdot (2 x^{4})] + \frac{d}{d x} [5]$

Step 2

Evaluate $\frac{d}{d x} [5 x^{5}]$ .

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Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{5}$ with respect to $x$ is $5 \frac{d}{d x} [x^{5}]$ .

$5 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [7 x^{2} \cdot (2 x^{4})] + \frac{d}{d x} [5]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$5 (5 x^{4}) + \frac{d}{d x} [7 x^{2} \cdot (2 x^{4})] + \frac{d}{d x} [5]$

Step 2.3

Multiply $5$ by $5$ .

$25 x^{4} + \frac{d}{d x} [7 x^{2} \cdot (2 x^{4})] + \frac{d}{d x} [5]$

Step 3

Evaluate $\frac{d}{d x} [7 x^{2} \cdot (2 x^{4})]$ .

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Step 3.1

Multiply $2$ by $7$ .

$25 x^{4} + \frac{d}{d x} [14 x^{2} \cdot x^{4}] + \frac{d}{d x} [5]$

Step 3.2

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

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Step 3.2.1

Move $x^{4}$ .

$25 x^{4} + \frac{d}{d x} [14 (x^{4} x^{2})] + \frac{d}{d x} [5]$

Step 3.2.2

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

$25 x^{4} + \frac{d}{d x} [14 x^{4 + 2}] + \frac{d}{d x} [5]$

Step 3.2.3

Add $4$ and $2$ .

$25 x^{4} + \frac{d}{d x} [14 x^{6}] + \frac{d}{d x} [5]$

Step 3.3

Since $14$ is constant with respect to $x$ , the derivative of $14 x^{6}$ with respect to $x$ is $14 \frac{d}{d x} [x^{6}]$ .

$25 x^{4} + 14 \frac{d}{d x} [x^{6}] + \frac{d}{d x} [5]$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$25 x^{4} + 14 (6 x^{5}) + \frac{d}{d x} [5]$

Step 3.5

Multiply $6$ by $14$ .

$25 x^{4} + 84 x^{5} + \frac{d}{d x} [5]$

Step 4

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$25 x^{4} + 84 x^{5} + 0$

Step 5

Simplify.

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Step 5.1

Add $25 x^{4} + 84 x^{5}$ and $0$ .

$25 x^{4} + 84 x^{5}$

Step 5.2

Reorder terms.

$84 x^{5} + 25 x^{4}$