Calculus Examples

$y = (2 - 5 x^{3}) \arcsin (x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ((2 - 5 x^{3}) \arcsin (x))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = 2 - 5 x^{3}$ and $g (x) = \arcsin (x)$ .

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

Step 3.2

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

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 3.3.3

Add $0$ and $\frac{d}{d x} [- 5 x^{3}]$ .

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

Step 3.3.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{3}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{3}]$ .

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

Step 3.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 3.3.6

Simplify the expression.

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Step 3.3.6.1

Multiply $3$ by $- 5$ .

$(2 - 5 x^{3}) \frac{1}{\sqrt{1 - x^{2}}} + \arcsin (x) (- 15 x^{2})$

Step 3.3.6.2

Reorder terms.

$(2 - 5 x^{3}) \frac{1}{\sqrt{1 - x^{2}}} - 15 x^{2} \arcsin (x)$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = (2 - 5 x^{3}) (\frac{1}{\sqrt{1 - x^{2}}}) - 15 x^{2} \arcsin (x)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = (2 - 5 x^{3}) (\frac{1}{\sqrt{1 - x^{2}}}) - 15 x^{2} \arcsin (x)$