Calculus Examples

$f (x) = 2 x^{e^{x}}$ $x = 1$

Step 1

Find the derivative.

Tap for more steps...

Step 1.1

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

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

Step 1.2

Use the properties of logarithms to simplify the differentiation.

Tap for more steps...

Step 1.2.1

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

$2 \frac{d}{d x} [e^{\ln (x^{e^{x}})}]$

Step 1.2.2

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

$2 \frac{d}{d x} [e^{e^{x} \ln (x)}]$

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

$2 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [e^{x} \ln (x)])$

Step 1.3.2

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

$2 (e^{u} \frac{d}{d x} [e^{x} \ln (x)])$

Step 1.3.3

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

$2 (e^{e^{x} \ln (x)} \frac{d}{d x} [e^{x} \ln (x)])$

Step 1.4

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 $g (x) = \ln (x)$ .

$2 e^{e^{x} \ln (x)} (e^{x} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [e^{x}])$

Step 1.5

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

$2 e^{e^{x} \ln (x)} (e^{x} \frac{1}{x} + \ln (x) \frac{d}{d x} [e^{x}])$

Step 1.6

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

$2 e^{e^{x} \ln (x)} (\frac{e^{x}}{x} + \ln (x) \frac{d}{d x} [e^{x}])$

Step 1.7

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

$2 e^{e^{x} \ln (x)} (\frac{e^{x}}{x} + \ln (x) e^{x})$

Step 1.8

Simplify.

Tap for more steps...

Step 1.8.1

Apply the distributive property.

$2 e^{e^{x} \ln (x)} \frac{e^{x}}{x} + 2 e^{e^{x} \ln (x)} (\ln (x) e^{x})$

Step 1.8.2

Combine terms.

Tap for more steps...

Step 1.8.2.1

Combine $\frac{e^{x}}{x}$ and $2$ .

$\frac{e^{x} \cdot 2}{x} e^{e^{x} \ln (x)} + 2 e^{e^{x} \ln (x)} (\ln (x) e^{x})$

Step 1.8.2.2

Combine $\frac{e^{x} \cdot 2}{x}$ and $e^{e^{x} \ln (x)}$ .

$\frac{e^{x} \cdot 2 e^{e^{x} \ln (x)}}{x} + 2 e^{e^{x} \ln (x)} (\ln (x) e^{x})$

Step 1.8.2.3

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

Tap for more steps...

Step 1.8.2.3.1

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

$\frac{e^{e^{x} \ln (x)} e^{x} \cdot 2}{x} + 2 e^{e^{x} \ln (x)} (\ln (x) e^{x})$

Step 1.8.2.3.2

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

$\frac{e^{e^{x} \ln (x) + x} \cdot 2}{x} + 2 e^{e^{x} \ln (x)} (\ln (x) e^{x})$

Step 1.8.2.4

Move $2$ to the left of $e^{e^{x} \ln (x) + x}$ .

$\frac{2 \cdot e^{e^{x} \ln (x) + x}}{x} + 2 e^{e^{x} \ln (x)} (\ln (x) e^{x})$

Step 1.8.2.5

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

$\frac{2 e^{e^{x} \ln (x) + x}}{x} + 2 e^{x + e^{x} \ln (x)} \ln (x)$

Step 1.8.2.6

To write $2 e^{x + e^{x} \ln (x)} \ln (x)$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{2 e^{e^{x} \ln (x) + x}}{x} + \frac{2 e^{x + e^{x} \ln (x)} \ln (x) x}{x}$

Step 1.8.2.7

Combine the numerators over the common denominator.

$\frac{2 e^{e^{x} \ln (x) + x} + 2 e^{x + e^{x} \ln (x)} \ln (x) x}{x}$

Step 1.8.3

Reorder terms.

$\frac{2 e^{x + e^{x} \ln (x)} x \ln (x) + 2 e^{e^{x} \ln (x) + x}}{x}$

Step 2

Replace the variable $x$ with $1$ in the expression.

$f' (1) = \frac{2 e^{(1) + e^{1} \ln (1)} (1) \ln (1) + 2 e^{e^{1} \ln (1) + 1}}{1}$

Step 3

Simplify the expression.

Tap for more steps...

Step 3.1

Remove parentheses.

$f' (1) = \frac{2 e^{(1) + e^{1} \ln (1)} (1) \ln (1) + 2 e^{e^{1} \ln (1) + 1}}{1}$

Step 3.2

Divide $2 e^{(1) + e^{1} \ln (1)} (1) \ln (1) + 2 e^{e^{1} \ln (1) + 1}$ by $1$ .

$f' (1) = 2 e^{(1) + e^{1} \ln (1)} (1) \ln (1) + 2 e^{e^{1} \ln (1) + 1}$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Simplify.

$f' (1) = 2 e^{1 + e \ln (1)} \cdot (1 \ln (1)) + 2 e^{e^{1} \ln (1) + 1}$

Step 4.1.2

The natural logarithm of $1$ is $0$ .

$f' (1) = 2 e^{1 + e \cdot 0} \cdot (1 \ln (1)) + 2 e^{e^{1} \ln (1) + 1}$

Step 4.1.3

Multiply $e$ by $0$ .

$f' (1) = 2 e^{1 + 0} \cdot (1 \ln (1)) + 2 e^{e^{1} \ln (1) + 1}$

Step 4.2

Add $1$ and $0$ .

$f' (1) = 2 e \cdot (1 \ln (1)) + 2 e^{e^{1} \ln (1) + 1}$

Step 4.3

Simplify.

$f' (1) = 2 e \cdot (1 \ln (1)) + 2 e^{e^{1} \ln (1) + 1}$

Step 4.4

Multiply $2$ by $1$ .

$f' (1) = 2 e \ln (1) + 2 e^{e^{1} \ln (1) + 1}$

Step 4.5

The natural logarithm of $1$ is $0$ .

$f' (1) = 2 e \cdot 0 + 2 e^{e^{1} \ln (1) + 1}$

Step 4.6

Multiply $2 e \cdot 0$ .

Tap for more steps...

Step 4.6.1

Multiply $0$ by $2$ .

$f' (1) = 0 e + 2 e^{e^{1} \ln (1) + 1}$

Step 4.6.2

Multiply $0$ by $e$ .

$f' (1) = 0 + 2 e^{e^{1} \ln (1) + 1}$

Step 4.7

Simplify each term.

Tap for more steps...

Step 4.7.1

Simplify.

$f' (1) = 0 + 2 e^{e \ln (1) + 1}$

Step 4.7.2

The natural logarithm of $1$ is $0$ .

$f' (1) = 0 + 2 e^{e \cdot 0 + 1}$

Step 4.7.3

Multiply $e$ by $0$ .

$f' (1) = 0 + 2 e^{0 + 1}$

Step 4.8

Add $0$ and $1$ .

$f' (1) = 0 + 2 e$

Step 4.9

Simplify.

$f' (1) = 0 + 2 e$

Step 5

Add $0$ and $2 e$ .

$f' (1) = 2 e$