Calculus Examples

$- 20 e^{1 - 4 x} + 20$

Step 1

By the Sum Rule, the derivative of $- 20 e^{1 - 4 x} + 20$ with respect to $x$ is $\frac{d}{d x} [- 20 e^{1 - 4 x}] + \frac{d}{d x} [20]$ .

$\frac{d}{d x} [- 20 e^{1 - 4 x}] + \frac{d}{d x} [20]$

Step 2

Evaluate $\frac{d}{d x} [- 20 e^{1 - 4 x}]$ .

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Step 2.1

Since $- 20$ is constant with respect to $x$ , the derivative of $- 20 e^{1 - 4 x}$ with respect to $x$ is $- 20 \frac{d}{d x} [e^{1 - 4 x}]$ .

$- 20 \frac{d}{d x} [e^{1 - 4 x}] + \frac{d}{d x} [20]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 1 - 4 x$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $1 - 4 x$ .

$- 20 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [1 - 4 x]) + \frac{d}{d x} [20]$

Step 2.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$- 20 (e^{u} \frac{d}{d x} [1 - 4 x]) + \frac{d}{d x} [20]$

Step 2.2.3

Replace all occurrences of $u$ with $1 - 4 x$ .

$- 20 (e^{1 - 4 x} \frac{d}{d x} [1 - 4 x]) + \frac{d}{d x} [20]$

Step 2.3

By the Sum Rule, the derivative of $1 - 4 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 4 x]$ .

$- 20 (e^{1 - 4 x} (\frac{d}{d x} [1] + \frac{d}{d x} [- 4 x])) + \frac{d}{d x} [20]$

Step 2.4

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

$- 20 (e^{1 - 4 x} (0 + \frac{d}{d x} [- 4 x])) + \frac{d}{d x} [20]$

Step 2.5

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$- 20 (e^{1 - 4 x} (0 - 4 \frac{d}{d x} [x])) + \frac{d}{d x} [20]$

Step 2.6

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

$- 20 (e^{1 - 4 x} (0 - 4 \cdot 1)) + \frac{d}{d x} [20]$

Step 2.7

Multiply $- 4$ by $1$ .

$- 20 (e^{1 - 4 x} (0 - 4)) + \frac{d}{d x} [20]$

Step 2.8

Subtract $4$ from $0$ .

$- 20 (e^{1 - 4 x} \cdot - 4) + \frac{d}{d x} [20]$

Step 2.9

Move $- 4$ to the left of $e^{1 - 4 x}$ .

$- 20 (- 4 \cdot e^{1 - 4 x}) + \frac{d}{d x} [20]$

Step 2.10

Multiply $- 4$ by $- 20$ .

$80 e^{1 - 4 x} + \frac{d}{d x} [20]$

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $20$ is constant with respect to $x$ , the derivative of $20$ with respect to $x$ is $0$ .

$80 e^{1 - 4 x} + 0$

Step 3.2

Add $80 e^{1 - 4 x}$ and $0$ .

$80 e^{1 - 4 x}$