Calculus Examples

$f (x) = (x^{3} + 7) e^{x}$

Step 1

Find the first derivative.

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Step 1.1

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 $f (x) = x^{3} + 7$ and $g (x) = e^{x}$ .

$(x^{3} + 7) \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{3} + 7]$

Step 1.2

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

$(x^{3} + 7) e^{x} + e^{x} \frac{d}{d x} [x^{3} + 7]$

Step 1.3

Differentiate.

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Step 1.3.1

By the Sum Rule, the derivative of $x^{3} + 7$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [7]$ .

$(x^{3} + 7) e^{x} + e^{x} (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [7])$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$(x^{3} + 7) e^{x} + e^{x} (3 x^{2} + \frac{d}{d x} [7])$

Step 1.3.3

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$(x^{3} + 7) e^{x} + e^{x} (3 x^{2} + 0)$

Step 1.3.4

Add $3 x^{2}$ and $0$ .

$(x^{3} + 7) e^{x} + e^{x} (3 x^{2})$

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

$x^{3} e^{x} + 7 e^{x} + e^{x} (3 x^{2})$

Step 1.4.2

Reorder terms.

$e^{x} x^{3} + 3 e^{x} x^{2} + 7 e^{x}$

Step 1.4.3

Reorder factors in $e^{x} x^{3} + 3 e^{x} x^{2} + 7 e^{x}$ .

$f' (x) = x^{3} e^{x} + 3 x^{2} e^{x} + 7 e^{x}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $x^{3} e^{x} + 3 x^{2} e^{x} + 7 e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{3} e^{x}] + \frac{d}{d x} [3 x^{2} e^{x}] + \frac{d}{d x} [7 e^{x}]$ .

$\frac{d}{d x} [x^{3} e^{x}] + \frac{d}{d x} [3 x^{2} e^{x}] + \frac{d}{d x} [7 e^{x}]$

Step 2.2

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

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Step 2.2.1

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 $f (x) = x^{3}$ and $g (x) = e^{x}$ .

$x^{3} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2} e^{x}] + \frac{d}{d x} [7 e^{x}]$

Step 2.2.2

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

$x^{3} e^{x} + e^{x} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2} e^{x}] + \frac{d}{d x} [7 e^{x}]$

Step 2.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$x^{3} e^{x} + e^{x} (3 x^{2}) + \frac{d}{d x} [3 x^{2} e^{x}] + \frac{d}{d x} [7 e^{x}]$

Step 2.3

Evaluate $\frac{d}{d x} [3 x^{2} e^{x}]$ .

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Step 2.3.1

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

$x^{3} e^{x} + e^{x} (3 x^{2}) + 3 \frac{d}{d x} [x^{2} e^{x}] + \frac{d}{d x} [7 e^{x}]$

Step 2.3.2

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 $f (x) = x^{2}$ and $g (x) = e^{x}$ .

$x^{3} e^{x} + e^{x} (3 x^{2}) + 3 (x^{2} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [7 e^{x}]$

Step 2.3.3

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

$x^{3} e^{x} + e^{x} (3 x^{2}) + 3 (x^{2} e^{x} + e^{x} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [7 e^{x}]$

Step 2.3.4

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

$x^{3} e^{x} + e^{x} (3 x^{2}) + 3 (x^{2} e^{x} + e^{x} (2 x)) + \frac{d}{d x} [7 e^{x}]$

Step 2.4

Evaluate $\frac{d}{d x} [7 e^{x}]$ .

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Step 2.4.1

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

$x^{3} e^{x} + e^{x} (3 x^{2}) + 3 (x^{2} e^{x} + e^{x} (2 x)) + 7 \frac{d}{d x} [e^{x}]$

Step 2.4.2

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

$x^{3} e^{x} + e^{x} (3 x^{2}) + 3 (x^{2} e^{x} + e^{x} (2 x)) + 7 e^{x}$

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

$x^{3} e^{x} + e^{x} (3 x^{2}) + 3 (x^{2} e^{x}) + 3 (e^{x} (2 x)) + 7 e^{x}$

Step 2.5.2

Combine terms.

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Step 2.5.2.1

Multiply $2$ by $3$ .

$x^{3} e^{x} + e^{x} (3 x^{2}) + 3 x^{2} e^{x} + 6 (e^{x} (x)) + 7 e^{x}$

Step 2.5.2.2

Add $e^{x} (3 x^{2})$ and $3 x^{2} e^{x}$ .

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Step 2.5.2.2.1

Move $e^{x}$ .

$x^{3} e^{x} + 3 x^{2} e^{x} + 3 x^{2} e^{x} + 6 e^{x} x + 7 e^{x}$

Step 2.5.2.2.2

Add $3 x^{2} e^{x}$ and $3 x^{2} e^{x}$ .

$x^{3} e^{x} + 6 x^{2} e^{x} + 6 e^{x} x + 7 e^{x}$

Step 2.5.3

Reorder terms.

$e^{x} x^{3} + 6 e^{x} x^{2} + 6 e^{x} x + 7 e^{x}$

Step 2.5.4

Reorder factors in $e^{x} x^{3} + 6 e^{x} x^{2} + 6 e^{x} x + 7 e^{x}$ .

$f'' (x) = x^{3} e^{x} + 6 x^{2} e^{x} + 6 x e^{x} + 7 e^{x}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $x^{3} e^{x} + 6 x^{2} e^{x} + 6 x e^{x} + 7 e^{x}$ .

$x^{3} e^{x} + 6 x^{2} e^{x} + 6 x e^{x} + 7 e^{x}$