Calculus Examples

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

Step 1

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

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

Step 2

Use the properties of logarithms to simplify the differentiation.

Tap for more steps...

Step 2.1

Rewrite $x^{3 e^{x}}$ as $e^{\ln (x^{3 e^{x}})}$ .

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

Step 2.2

Expand $\ln (x^{3 e^{x}})$ by moving $3 e^{x}$ outside the logarithm.

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

Step 3

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 $g (x) = 3 e^{x} \ln (x)$ .

Tap for more steps...

Step 3.1

To apply the Chain Rule, set $u$ as $3 e^{x} \ln (x)$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $3 e^{x} \ln (x)$ .

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

Step 4

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 4.1

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

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

Step 4.2

Multiply $3$ by $5$ .

$15 e^{3 e^{x} \ln (x)} \frac{d}{d x} [e^{x} \ln (x)]$

Step 5

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = e^{x}$ and $g (x) = \ln (x)$ .

$15 e^{3 e^{x} \ln (x)} (e^{x} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [e^{x}])$

Step 6

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$15 e^{3 e^{x} \ln (x)} (e^{x} \frac{1}{x} + \ln (x) \frac{d}{d x} [e^{x}])$

Step 7

Combine $e^{x}$ and $\frac{1}{x}$ .

$15 e^{3 e^{x} \ln (x)} (\frac{e^{x}}{x} + \ln (x) \frac{d}{d x} [e^{x}])$

Step 8

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

$15 e^{3 e^{x} \ln (x)} (\frac{e^{x}}{x} + \ln (x) e^{x})$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Apply the distributive property.

$15 e^{3 e^{x} \ln (x)} \frac{e^{x}}{x} + 15 e^{3 e^{x} \ln (x)} (\ln (x) e^{x})$

Step 9.2

Combine terms.

Tap for more steps...

Step 9.2.1

Combine $\frac{e^{x}}{x}$ and $15$ .

$\frac{e^{x} \cdot 15}{x} e^{3 e^{x} \ln (x)} + 15 e^{3 e^{x} \ln (x)} (\ln (x) e^{x})$

Step 9.2.2

Combine $\frac{e^{x} \cdot 15}{x}$ and $e^{3 e^{x} \ln (x)}$ .

$\frac{e^{x} \cdot 15 e^{3 e^{x} \ln (x)}}{x} + 15 e^{3 e^{x} \ln (x)} (\ln (x) e^{x})$

Step 9.2.3

Multiply $e^{x}$ by $e^{3 e^{x} \ln (x)}$ by adding the exponents.

Tap for more steps...

Step 9.2.3.1

Move $e^{3 e^{x} \ln (x)}$ .

$\frac{e^{3 e^{x} \ln (x)} e^{x} \cdot 15}{x} + 15 e^{3 e^{x} \ln (x)} (\ln (x) e^{x})$

Step 9.2.3.2

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

$\frac{e^{3 e^{x} \ln (x) + x} \cdot 15}{x} + 15 e^{3 e^{x} \ln (x)} (\ln (x) e^{x})$

Step 9.2.3.3

Multiply $3 e^{x} \ln (x)$ .

Tap for more steps...

Step 9.2.3.3.1

Reorder $\ln (x)$ and $3$ .

$\frac{e^{3 \cdot \ln (x) e^{x} + x} \cdot 15}{x} + 15 e^{3 e^{x} \ln (x)} (\ln (x) e^{x})$

Step 9.2.3.3.2

Simplify $3 \cdot \ln (x)$ by moving $3$ inside the logarithm.

$\frac{e^{\ln (x^{3}) e^{x} + x} \cdot 15}{x} + 15 e^{3 e^{x} \ln (x)} (\ln (x) e^{x})$

Step 9.2.3.4

Reorder factors in $\ln (x^{3}) e^{x} + x$ .

$\frac{e^{e^{x} \ln (x^{3}) + x} \cdot 15}{x} + 15 e^{3 e^{x} \ln (x)} (\ln (x) e^{x})$

Step 9.2.4

Move $15$ to the left of $e^{e^{x} \ln (x^{3}) + x}$ .

$\frac{15 \cdot e^{e^{x} \ln (x^{3}) + x}}{x} + 15 e^{3 e^{x} \ln (x)} (\ln (x) e^{x})$

Step 9.2.5

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

$\frac{15 e^{e^{x} \ln (x^{3}) + x}}{x} + 15 e^{x + 3 e^{x} \ln (x)} \ln (x)$

Step 9.2.6

To write $15 e^{x + 3 e^{x} \ln (x)} \ln (x)$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{15 e^{e^{x} \ln (x^{3}) + x}}{x} + \frac{15 e^{x + 3 e^{x} \ln (x)} \ln (x) x}{x}$

Step 9.2.7

Combine the numerators over the common denominator.

$\frac{15 e^{e^{x} \ln (x^{3}) + x} + 15 e^{x + 3 e^{x} \ln (x)} \ln (x) x}{x}$

Step 9.3

Reorder terms.

$\frac{15 e^{x + 3 e^{x} \ln (x)} x \ln (x) + 15 e^{e^{x} \ln (x^{3}) + x}}{x}$