Calculus Examples

$y = 2 e^{4 x^{5}} - 2 x^{3} + 4$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $2 e^{4 x^{5}} - 2 x^{3} + 4$ with respect to $e$ is $\frac{d}{d e} [2 e^{4 x^{5}}] + \frac{d}{d e} [- 2 x^{3}] + \frac{d}{d e} [4]$ .

$\frac{d}{d e} [2 e^{4 x^{5}}] + \frac{d}{d e} [- 2 x^{3}] + \frac{d}{d e} [4]$

Step 1.2

Since $2 e^{4 x^{5}}$ is constant with respect to $e$ , the derivative of $2 e^{4 x^{5}}$ with respect to $e$ is $0$ .

$0 + \frac{d}{d e} [- 2 x^{3}] + \frac{d}{d e} [4]$

Step 1.3

Since $- 2 x^{3}$ is constant with respect to $e$ , the derivative of $- 2 x^{3}$ with respect to $e$ is $0$ .

$0 + 0 + \frac{d}{d e} [4]$

Step 1.4

Since $4$ is constant with respect to $e$ , the derivative of $4$ with respect to $e$ is $0$ .

$0 + 0 + 0$

Step 1.5

Combine terms.

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Step 1.5.1

Add $0$ and $0$ .

$0 + 0$

Step 1.5.2

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 $0$ .

$f'' (e) = 0$

Step 3

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

$f''' (e) = 0$