Calculus Examples

$f (x) = \ln (x) \cdot \arcsin (x) - \frac{e^{x}}{x} + 1$

Step 1

By the Sum Rule, the derivative of $\ln (x) \cdot \arcsin (x) - \frac{e^{x}}{x} + 1$ with respect to $x$ is $\frac{d}{d x} [\ln (x) \cdot \arcsin (x)] + \frac{d}{d x} [- \frac{e^{x}}{x}] + \frac{d}{d x} [1]$ .

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

Step 2

Evaluate $\frac{d}{d x} [\ln (x) \cdot \arcsin (x)]$ .

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Step 2.1

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 $f (x) = \ln (x)$ and $g (x) = \arcsin (x)$ .

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

Step 2.2

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

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

Step 2.3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

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

Step 2.4

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

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

Step 2.5

Combine $\arcsin (x)$ and $\frac{1}{x}$ .

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

Step 3

Evaluate $\frac{d}{d x} [- \frac{e^{x}}{x}]$ .

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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 $f (x) = - 1$ and $g (x) = \frac{e^{x}}{x}$ .

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

Step 3.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = e^{x}$ and $g (x) = x$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{\ln (x)}{\sqrt{1 - x^{2}}} + \frac{\arcsin (x)}{x} - \frac{x e^{x} - e^{x} \cdot 1}{x^{2}} + \frac{e^{x}}{x} \cdot 0 + \frac{d}{d x} [1]$

Step 3.6

Multiply $- 1$ by $1$ .

$\frac{\ln (x)}{\sqrt{1 - x^{2}}} + \frac{\arcsin (x)}{x} - \frac{x e^{x} - e^{x}}{x^{2}} + \frac{e^{x}}{x} \cdot 0 + \frac{d}{d x} [1]$

Step 3.7

Multiply $\frac{e^{x}}{x}$ by $0$ .

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

Step 3.8

Add $- \frac{x e^{x} - e^{x}}{x^{2}}$ and $0$ .

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

Step 4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{\ln (x)}{\sqrt{1 - x^{2}}} + \frac{\arcsin (x)}{x} - \frac{x e^{x} - e^{x}}{x^{2}} + 0$

Step 5

Simplify.

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Step 5.1

Add $\frac{\ln (x)}{\sqrt{1 - x^{2}}} + \frac{\arcsin (x)}{x} - \frac{x e^{x} - e^{x}}{x^{2}}$ and $0$ .

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

Step 5.2

Reorder terms.

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