Calculus Examples

$y = 4 x e^{x}$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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 $f (x) = x$ and $g (x) = e^{x}$ .

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

Step 1.3

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

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

Step 1.4

Differentiate using the Power Rule.

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Step 1.4.1

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

$4 (x e^{x} + e^{x} \cdot 1)$

Step 1.4.2

Multiply $e^{x}$ by $1$ .

$4 (x e^{x} + e^{x})$

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

$4 (x e^{x}) + 4 e^{x}$

Step 1.5.2

Reorder terms.

$4 e^{x} x + 4 e^{x}$

Step 1.5.3

Reorder factors in $4 e^{x} x + 4 e^{x}$ .

$f' (x) = 4 x e^{x} + 4 e^{x}$

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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 $f (x) = x$ and $g (x) = e^{x}$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $e^{x}$ by $1$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

$4 (x e^{x} + e^{x}) + 4 e^{x}$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$4 (x e^{x}) + 4 e^{x} + 4 e^{x}$

Step 2.4.2

Add $4 e^{x}$ and $4 e^{x}$ .

$4 x e^{x} + 8 e^{x}$

Step 2.4.3

Reorder terms.

$4 e^{x} x + 8 e^{x}$

Step 2.4.4

Reorder factors in $4 e^{x} x + 8 e^{x}$ .

$f'' (x) = 4 x e^{x} + 8 e^{x}$