Calculus Examples

$- (x + 1) \sin (\frac{x^{2}}{2})$

Step 1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (x + 1) \sin (\frac{x^{2}}{2})$ with respect to $x$ is $- \frac{d}{d x} [(x + 1) \sin (\frac{x^{2}}{2})]$ .

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

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 $f (x) = x + 1$ and $g (x) = \sin (\frac{x^{2}}{2})$ .

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = \frac{x^{2}}{2}$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $\frac{x^{2}}{2}$ .

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

Step 3.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$- ((x + 1) (\cos (u) \frac{d}{d x} [\frac{x^{2}}{2}]) + \sin (\frac{x^{2}}{2}) \frac{d}{d x} [x + 1])$

Step 3.3

Replace all occurrences of $u$ with $\frac{x^{2}}{2}$ .

$- ((x + 1) (\cos (\frac{x^{2}}{2}) \frac{d}{d x} [\frac{x^{2}}{2}]) + \sin (\frac{x^{2}}{2}) \frac{d}{d x} [x + 1])$

Step 4

Differentiate.

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Step 4.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{2}]$ .

$- ((x + 1) \cos (\frac{x^{2}}{2}) (\frac{1}{2} \frac{d}{d x} [x^{2}]) + \sin (\frac{x^{2}}{2}) \frac{d}{d x} [x + 1])$

Step 4.2

Combine $\frac{1}{2}$ and $\cos (\frac{x^{2}}{2})$ .

$- ((x + 1) \frac{\cos (\frac{x^{2}}{2})}{2} \frac{d}{d x} [x^{2}] + \sin (\frac{x^{2}}{2}) \frac{d}{d x} [x + 1])$

Step 4.3

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

$- ((x + 1) \frac{\cos (\frac{x^{2}}{2})}{2} (2 x) + \sin (\frac{x^{2}}{2}) \frac{d}{d x} [x + 1])$

Step 4.4

Simplify terms.

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Step 4.4.1

Combine $2$ and $\frac{\cos (\frac{x^{2}}{2})}{2}$ .

$- ((x + 1) \frac{2 \cos (\frac{x^{2}}{2})}{2} x + \sin (\frac{x^{2}}{2}) \frac{d}{d x} [x + 1])$

Step 4.4.2

Combine $x$ and $\frac{2 \cos (\frac{x^{2}}{2})}{2}$ .

$- ((x + 1) \frac{x (2 \cos (\frac{x^{2}}{2}))}{2} + \sin (\frac{x^{2}}{2}) \frac{d}{d x} [x + 1])$

Step 4.4.3

Cancel the common factor of $2$ .

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Step 4.4.3.1

Cancel the common factor.

$- ((x + 1) \frac{x (2 \cos (\frac{x^{2}}{2}))}{2} + \sin (\frac{x^{2}}{2}) \frac{d}{d x} [x + 1])$

Step 4.4.3.2

Divide $x (\cos (\frac{x^{2}}{2}))$ by $1$ .

$- ((x + 1) (x (\cos (\frac{x^{2}}{2}))) + \sin (\frac{x^{2}}{2}) \frac{d}{d x} [x + 1])$

Step 4.5

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$- ((x + 1) x \cos (\frac{x^{2}}{2}) + \sin (\frac{x^{2}}{2}) (\frac{d}{d x} [x] + \frac{d}{d x} [1]))$

Step 4.6

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

$- ((x + 1) x \cos (\frac{x^{2}}{2}) + \sin (\frac{x^{2}}{2}) (1 + \frac{d}{d x} [1]))$

Step 4.7

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

$- ((x + 1) x \cos (\frac{x^{2}}{2}) + \sin (\frac{x^{2}}{2}) (1 + 0))$

Step 4.8

Simplify the expression.

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Step 4.8.1

Add $1$ and $0$ .

$- ((x + 1) x \cos (\frac{x^{2}}{2}) + \sin (\frac{x^{2}}{2}) \cdot 1)$

Step 4.8.2

Multiply $\sin (\frac{x^{2}}{2})$ by $1$ .

$- ((x + 1) x \cos (\frac{x^{2}}{2}) + \sin (\frac{x^{2}}{2}))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$- ((x \cdot x + 1 x) \cos (\frac{x^{2}}{2}) + \sin (\frac{x^{2}}{2}))$

Step 5.2

Apply the distributive property.

$- (x \cdot x \cos (\frac{x^{2}}{2}) + 1 x \cos (\frac{x^{2}}{2}) + \sin (\frac{x^{2}}{2}))$

Step 5.3

Apply the distributive property.

$- (x \cdot x \cos (\frac{x^{2}}{2})) - (1 x \cos (\frac{x^{2}}{2})) - \sin (\frac{x^{2}}{2})$

Step 5.4

Combine terms.

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Step 5.4.1

Raise $x$ to the power of $1$ .

$- (x^{1} x \cos (\frac{x^{2}}{2})) - (1 x \cos (\frac{x^{2}}{2})) - \sin (\frac{x^{2}}{2})$

Step 5.4.2

Raise $x$ to the power of $1$ .

$- (x^{1} x^{1} \cos (\frac{x^{2}}{2})) - (1 x \cos (\frac{x^{2}}{2})) - \sin (\frac{x^{2}}{2})$

Step 5.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- (x^{1 + 1} \cos (\frac{x^{2}}{2})) - (1 x \cos (\frac{x^{2}}{2})) - \sin (\frac{x^{2}}{2})$

Step 5.4.4

Add $1$ and $1$ .

$- (x^{2} \cos (\frac{x^{2}}{2})) - (1 x \cos (\frac{x^{2}}{2})) - \sin (\frac{x^{2}}{2})$

Step 5.4.5

Multiply $x$ by $1$ .

$- x^{2} \cos (\frac{x^{2}}{2}) - x \cos (\frac{x^{2}}{2}) - \sin (\frac{x^{2}}{2})$