Calculus Examples

$f (x) = x^{2} \cos (x)$

Step 1

Find the first derivative.

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Step 1.1

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 $g (x) = \cos (x)$ .

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

Step 1.2

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

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

Step 1.3

Differentiate using the Power Rule.

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Step 1.3.1

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

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

Step 1.3.2

Reorder terms.

$f' (x) = - x^{2} \sin (x) + 2 x \cos (x)$

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

Evaluate $\frac{d}{d x} [- x^{2} \sin (x)]$ .

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.3

Evaluate $\frac{d}{d x} [2 x \cos (x)]$ .

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Step 2.3.1

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

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

Step 2.3.2

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 $g (x) = \cos (x)$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Apply the distributive property.

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

Step 2.4.3

Combine terms.

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Step 2.4.3.1

Multiply $2$ by $- 1$ .

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

Step 2.4.3.2

Multiply $- 1$ by $2$ .

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

Step 2.4.3.3

Subtract $2 x \sin (x)$ from $- 2 \sin (x) x$ .

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Step 2.4.3.3.1

Move $\sin (x)$ .

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

Step 2.4.3.3.2

Subtract $2 x \sin (x)$ from $- 2 x \sin (x)$ .

$f'' (x) = - x^{2} \cos (x) - 4 x \sin (x) + 2 \cos (x)$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- x^{2} \cos (x) - 4 x \sin (x) + 2 \cos (x)$ .

$- x^{2} \cos (x) - 4 x \sin (x) + 2 \cos (x)$