Calculus Examples

$f (x) = e^{x} + 100 e + 8 x^{4} + 49$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $e^{x} + 100 e + 8 x^{4} + 49$ with respect to $x$ is $\frac{d}{d x} [e^{x}] + \frac{d}{d x} [100 e] + \frac{d}{d x} [8 x^{4}] + \frac{d}{d x} [49]$ .

$\frac{d}{d x} [e^{x}] + \frac{d}{d x} [100 e] + \frac{d}{d x} [8 x^{4}] + \frac{d}{d x} [49]$

Step 1.2

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

$e^{x} + \frac{d}{d x} [100 e] + \frac{d}{d x} [8 x^{4}] + \frac{d}{d x} [49]$

Step 1.3

Since $100 e$ is constant with respect to $x$ , the derivative of $100 e$ with respect to $x$ is $0$ .

$e^{x} + 0 + \frac{d}{d x} [8 x^{4}] + \frac{d}{d x} [49]$

Step 1.4

Evaluate $\frac{d}{d x} [8 x^{4}]$ .

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Step 1.4.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{4}$ with respect to $x$ is $8 \frac{d}{d x} [x^{4}]$ .

$e^{x} + 0 + 8 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [49]$

Step 1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$e^{x} + 0 + 8 (4 x^{3}) + \frac{d}{d x} [49]$

Step 1.4.3

Multiply $4$ by $8$ .

$e^{x} + 0 + 32 x^{3} + \frac{d}{d x} [49]$

Step 1.5

Since $49$ is constant with respect to $x$ , the derivative of $49$ with respect to $x$ is $0$ .

$e^{x} + 0 + 32 x^{3} + 0$

Step 1.6

Simplify.

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Step 1.6.1

Combine terms.

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Step 1.6.1.1

Add $e^{x}$ and $0$ .

$e^{x} + 32 x^{3} + 0$

Step 1.6.1.2

Add $e^{x} + 32 x^{3}$ and $0$ .

$e^{x} + 32 x^{3}$

Step 1.6.2

Reorder terms.

$f' (x) = 32 x^{3} + e^{x}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $32 x^{3} + e^{x}$ with respect to $x$ is $\frac{d}{d x} [32 x^{3}] + \frac{d}{d x} [e^{x}]$ .

$\frac{d}{d x} [32 x^{3}] + \frac{d}{d x} [e^{x}]$

Step 2.2

Evaluate $\frac{d}{d x} [32 x^{3}]$ .

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Step 2.2.1

Since $32$ is constant with respect to $x$ , the derivative of $32 x^{3}$ with respect to $x$ is $32 \frac{d}{d x} [x^{3}]$ .

$32 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [e^{x}]$

Step 2.2.2

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

$32 (3 x^{2}) + \frac{d}{d x} [e^{x}]$

Step 2.2.3

Multiply $3$ by $32$ .

$96 x^{2} + \frac{d}{d x} [e^{x}]$

Step 2.3

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

$f'' (x) = 96 x^{2} + e^{x}$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $96 x^{2} + e^{x}$ .

$96 x^{2} + e^{x}$