Calculus Examples

$y = \arccos (3 x - 5)$

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 $g (x) = 3 x - 5$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $3 x - 5$ .

$\frac{d}{d u} [\arccos (u)] \frac{d}{d x} [3 x - 5]$

Step 1.2

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

$- \frac{1}{\sqrt{1 - u^{2}}} \frac{d}{d x} [3 x - 5]$

Step 1.3

Replace all occurrences of $u$ with $3 x - 5$ .

$- \frac{1}{\sqrt{1 - {(3 x - 5)}^{2}}} \frac{d}{d x} [3 x - 5]$

Step 2

Differentiate.

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Step 2.1

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

$- \frac{1}{\sqrt{1 - {(3 x - 5)}^{2}}} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 5])$

Step 2.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$- \frac{1}{\sqrt{1 - {(3 x - 5)}^{2}}} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 5])$

Step 2.3

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

$- \frac{1}{\sqrt{1 - {(3 x - 5)}^{2}}} (3 \cdot 1 + \frac{d}{d x} [- 5])$

Step 2.4

Multiply $3$ by $1$ .

$- \frac{1}{\sqrt{1 - {(3 x - 5)}^{2}}} (3 + \frac{d}{d x} [- 5])$

Step 2.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$- \frac{1}{\sqrt{1 - {(3 x - 5)}^{2}}} (3 + 0)$

Step 2.6

Combine fractions.

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Step 2.6.1

Add $3$ and $0$ .

$- \frac{1}{\sqrt{1 - {(3 x - 5)}^{2}}} \cdot 3$

Step 2.6.2

Multiply $3$ by $- 1$ .

$- 3 \frac{1}{\sqrt{1 - {(3 x - 5)}^{2}}}$

Step 2.6.3

Combine $- 3$ and $\frac{1}{\sqrt{1 - {(3 x - 5)}^{2}}}$ .

$\frac{- 3}{\sqrt{1 - {(3 x - 5)}^{2}}}$

Step 2.6.4

Move the negative in front of the fraction.

$- \frac{3}{\sqrt{1 - {(3 x - 5)}^{2}}}$