Calculus Examples

$f (x)$

Step 1

Differentiate using the function rule which states that the derivative of $f (x)$ is $\frac{d f}{d x}$ .

$f' (f) = \frac{d f}{d x}$

Step 2

Find the second derivative.

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Step 2.1

Cancel the common factor of $d$ .

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Step 2.1.1

Cancel the common factor.

$f'' (f) = \frac{d}{d f} (\frac{d f}{d x})$

Step 2.1.2

Rewrite the expression.

$f'' (f) = \frac{d}{d f} (\frac{f}{x})$

Step 2.2

Since $\frac{1}{x}$ is constant with respect to $f$ , the derivative of $\frac{f}{x}$ with respect to $f$ is $\frac{1}{x} \frac{d}{d f} [f]$ .

$f'' (f) = \frac{1}{x} \cdot \frac{d}{d f} (f)$

Step 2.3

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

$f'' (f) = \frac{1}{x} \cdot 1$

Step 2.4

Multiply $\frac{1}{x}$ by $1$ .

$f'' (f) = \frac{1}{x}$

Step 3

Since $\frac{1}{x}$ is constant with respect to $f$ , the derivative of $\frac{1}{x}$ with respect to $f$ is $0$ .

$f''' (f) = 0$

Step 4

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

$f^{4} (f) = 0$