Calculus Examples

$f (x) = - 4 e^{7 x}$

Step 1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 e^{7 x}$ with respect to $x$ is $- 4 \frac{d}{d x} [e^{7 x}]$ .

$- 4 \frac{d}{d x} [e^{7 x}]$

Step 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 $g (x) = 7 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $7 x$ .

$- 4 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [7 x])$

Step 2.2

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

$- 4 (e^{u} \frac{d}{d x} [7 x])$

Step 2.3

Replace all occurrences of $u$ with $7 x$ .

$- 4 (e^{7 x} \frac{d}{d x} [7 x])$

Step 3

Differentiate.

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Step 3.1

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

$- 4 e^{7 x} (7 \frac{d}{d x} [x])$

Step 3.2

Multiply $7$ by $- 4$ .

$- 28 e^{7 x} \frac{d}{d x} [x]$

Step 3.3

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

$- 28 e^{7 x} \cdot 1$

Step 3.4

Multiply $- 28$ by $1$ .

$- 28 e^{7 x}$