Calculus Examples

$f = {(x)}^{2} + 20 x + 400$

Step 1

Remove parentheses.

$f = x^{2} + 20 x + 400$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (f) = \frac{d}{d x} (x^{2} + 20 x + 400)$

Step 3

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

$f'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate.

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Step 4.1.1

By the Sum Rule, the derivative of $x^{2} + 20 x + 400$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [20 x] + \frac{d}{d x} [400]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [20 x] + \frac{d}{d x} [400]$

Step 4.1.2

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

$2 x + \frac{d}{d x} [20 x] + \frac{d}{d x} [400]$

Step 4.2

Evaluate $\frac{d}{d x} [20 x]$ .

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Step 4.2.1

Since $20$ is constant with respect to $x$ , the derivative of $20 x$ with respect to $x$ is $20 \frac{d}{d x} [x]$ .

$2 x + 20 \frac{d}{d x} [x] + \frac{d}{d x} [400]$

Step 4.2.2

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

$2 x + 20 \cdot 1 + \frac{d}{d x} [400]$

Step 4.2.3

Multiply $20$ by $1$ .

$2 x + 20 + \frac{d}{d x} [400]$

Step 4.3

Differentiate using the Constant Rule.

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Step 4.3.1

Since $400$ is constant with respect to $x$ , the derivative of $400$ with respect to $x$ is $0$ .

$2 x + 20 + 0$

Step 4.3.2

Add $2 x + 20$ and $0$ .

$2 x + 20$

Step 5

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

$f' = 2 x + 20$

Step 6

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

$\frac{d f}{d x} = 2 x + 20$