Calculus Examples

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

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

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

Step 1.2

Since $e^{x}$ is constant with respect to $e$ , the derivative of $e^{x}$ with respect to $e$ is $0$ .

$0 + \frac{d}{d e} [100 e] + \frac{d}{d e} [8 x^{4}] + \frac{d}{d e} [49]$

Step 1.3

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

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

Step 1.4

Since $8 x^{4}$ is constant with respect to $e$ , the derivative of $8 x^{4}$ with respect to $e$ is $0$ .

$0 + 0 + 0 + \frac{d}{d e} [49]$

Step 1.5

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

$0 + 0 + 0 + 0$

Step 1.6

Combine terms.

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Step 1.6.1

Add $0$ and $0$ .

$0 + 0 + 0$

Step 1.6.2

Add $0$ and $0$ .

$0 + 0$

Step 1.6.3

Add $0$ and $0$ .

$f' (e) = 0$

Step 2

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

$f'' (e) = 0$