Calculus Examples

$f (x) = 5 e^{3 x^{2} - 5 x}$

Step 1

Since $5$ is constant with respect to $x$ , the derivative of $5 e^{3 x^{2} - 5 x}$ with respect to $x$ is $5 \frac{d}{d x} [e^{3 x^{2} - 5 x}]$ .

$5 \frac{d}{d x} [e^{3 x^{2} - 5 x}]$

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 $g (x) = 3 x^{2} - 5 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 x^{2} - 5 x$ .

$5 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [3 x^{2} - 5 x])$

Step 2.2

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

$5 (e^{u} \frac{d}{d x} [3 x^{2} - 5 x])$

Step 2.3

Replace all occurrences of $u$ with $3 x^{2} - 5 x$ .

$5 (e^{3 x^{2} - 5 x} \frac{d}{d x} [3 x^{2} - 5 x])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $3 x^{2} - 5 x$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 5 x]$ .

$5 e^{3 x^{2} - 5 x} (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 5 x])$

Step 3.2

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

$5 e^{3 x^{2} - 5 x} (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x])$

Step 3.3

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

$5 e^{3 x^{2} - 5 x} (3 (2 x) + \frac{d}{d x} [- 5 x])$

Step 3.4

Multiply $2$ by $3$ .

$5 e^{3 x^{2} - 5 x} (6 x + \frac{d}{d x} [- 5 x])$

Step 3.5

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

$5 e^{3 x^{2} - 5 x} (6 x - 5 \frac{d}{d x} [x])$

Step 3.6

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

$5 e^{3 x^{2} - 5 x} (6 x - 5 \cdot 1)$

Step 3.7

Multiply $- 5$ by $1$ .

$5 e^{3 x^{2} - 5 x} (6 x - 5)$