Calculus Examples

$\arccos (- \frac{x}{5})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arccos (x)$ and $g (x) = - \frac{x}{5}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $- \frac{x}{5}$ .

$\frac{d}{d u} [\arccos (u)] \frac{d}{d x} [- \frac{x}{5}]$

Step 1.2

The derivative of $\arccos (u)$ with respect to $u$ is $- \frac{1}{\sqrt{1 - u^{2}}}$ .

$- \frac{1}{\sqrt{1 - u^{2}}} \frac{d}{d x} [- \frac{x}{5}]$

Step 1.3

Replace all occurrences of $u$ with $- \frac{x}{5}$ .

$- \frac{1}{\sqrt{1 - {(- \frac{x}{5})}^{2}}} \frac{d}{d x} [- \frac{x}{5}]$

Step 2

Differentiate.

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Step 2.1

Factor $- 1$ out of $- \frac{x}{5}$ .

$- \frac{1}{\sqrt{1 - {(- (\frac{x}{5}))}^{2}}} \frac{d}{d x} [- \frac{x}{5}]$

Step 2.2

Simplify the expression.

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Step 2.2.1

Apply the product rule to $- (\frac{x}{5})$ .

$- \frac{1}{\sqrt{1 - ({(- 1)}^{2} {(\frac{x}{5})}^{2})}} \frac{d}{d x} [- \frac{x}{5}]$

Step 2.2.2

Raise $- 1$ to the power of $2$ .

$- \frac{1}{\sqrt{1 - (1 {(\frac{x}{5})}^{2})}} \frac{d}{d x} [- \frac{x}{5}]$

Step 2.2.3

Multiply ${(\frac{x}{5})}^{2}$ by $1$ .

$- \frac{1}{\sqrt{1 - {(\frac{x}{5})}^{2}}} \frac{d}{d x} [- \frac{x}{5}]$

Step 2.3

Since $- \frac{1}{5}$ is constant with respect to $x$ , the derivative of $- \frac{x}{5}$ with respect to $x$ is $- \frac{1}{5} \frac{d}{d x} [x]$ .

$- \frac{1}{\sqrt{1 - {(\frac{x}{5})}^{2}}} (- \frac{1}{5} \frac{d}{d x} [x])$

Step 2.4

Combine fractions.

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Step 2.4.1

Multiply $- 1$ by $- 1$ .

$1 \frac{1}{\sqrt{1 - {(\frac{x}{5})}^{2}}} (\frac{1}{5} \frac{d}{d x} [x])$

Step 2.4.2

Multiply $\frac{1}{\sqrt{1 - {(\frac{x}{5})}^{2}}}$ by $1$ .

$\frac{1}{\sqrt{1 - {(\frac{x}{5})}^{2}}} (\frac{1}{5} \frac{d}{d x} [x])$

Step 2.4.3

Multiply $\frac{1}{5}$ by $\frac{1}{\sqrt{1 - {(\frac{x}{5})}^{2}}}$ .

$\frac{1}{5 \sqrt{1 - {(\frac{x}{5})}^{2}}} \frac{d}{d x} [x]$

Step 2.5

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

$\frac{1}{5 \sqrt{1 - {(\frac{x}{5})}^{2}}} \cdot 1$

Step 2.6

Multiply $\frac{1}{5 \sqrt{1 - {(\frac{x}{5})}^{2}}}$ by $1$ .

$\frac{1}{5 \sqrt{1 - {(\frac{x}{5})}^{2}}}$

Step 3

Simplify.

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Step 3.1

Apply the product rule to $\frac{x}{5}$ .

$\frac{1}{5 \sqrt{1 - \frac{x^{2}}{5^{2}}}}$

Step 3.2

Raise $5$ to the power of $2$ .

$\frac{1}{5 \sqrt{1 - \frac{x^{2}}{25}}}$