Calculus Examples

$f (x) = 3 x \arccos (x)$

Step 1

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

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \arccos (x)$ .

$3 (x \frac{d}{d x} [\arccos (x)] + \arccos (x) \frac{d}{d x} [x])$

Step 3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

Combine $x$ and $\frac{1}{\sqrt{1 - x^{2}}}$ .

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

Step 4.2

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

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

Step 4.3

Multiply $\arccos (x)$ by $1$ .

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Combine terms.

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Step 5.2.1

Multiply $- 1$ by $3$ .

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

Step 5.2.2

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

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

Step 5.2.3

Move the negative in front of the fraction.

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