Calculus Examples

$e^{x} + 100 e + 8 x^{4} + 49$ at $x = 1$

Step 1

Write $e^{x} + 100 e + 8 x^{4} + 49$ as an equation.

$y = e^{x} + 100 e + 8 x^{4} + 49$

Step 2

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 3

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 4

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 5

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

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Step 5.1

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 5.2

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 5.3

Multiply $4$ by $8$ .

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

Step 6

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 7

Simplify.

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Step 7.1

Combine terms.

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.2

Reorder terms.

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

Step 8

Evaluate the derivative at $x = 1$ .

$32 {(1)}^{3} + e^{1}$

Step 9

Simplify each term.

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Step 9.1

One to any power is one.

$32 \cdot 1 + e^{1}$

Step 9.2

Multiply $32$ by $1$ .

$32 + e^{1}$

Step 9.3

Simplify.

$32 + e$