Calculus Examples

$y = x^{2} \arcsin (e^{x})$

Step 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 $f (x) = x^{2}$ and $g (x) = \arcsin (e^{x})$ .

$x^{2} \frac{d}{d x} [\arcsin (e^{x})] + \arcsin (e^{x}) \frac{d}{d x} [x^{2}]$

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $e^{x}$ .

$x^{2} (\frac{d}{d u} [\arcsin (u)] \frac{d}{d x} [e^{x}]) + \arcsin (e^{x}) \frac{d}{d x} [x^{2}]$

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $e^{x}$ .

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

Step 3

Combine fractions.

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Step 3.1

Multiply the exponents in ${(e^{x})}^{2}$ .

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Step 3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.1.2

Move $2$ to the left of $x$ .

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

Step 3.2

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

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

Step 4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

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

Step 5

Differentiate using the Power Rule.

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Step 5.1

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

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

Step 5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 5.3

Reorder terms.

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