Calculus Examples

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

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

$f' (x) = x^{2} \frac{d}{d x} (\sin (x)) + \sin (x) \frac{d}{d x} (x^{2})$

Step 1.2

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

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

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 $n = 2$ .

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

Step 1.3.2

Reorder terms.

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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

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Step 2.3.1

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

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

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Add $\cos (x) (2 x)$ and $2 x \cos (x)$ .

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Step 2.4.2.1

Move $\cos (x)$ .

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

Step 2.4.2.2

Add $2 x \cos (x)$ and $2 x \cos (x)$ .

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

Step 2.4.3

Reorder terms.

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

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $- x^{2} \sin (x) + 4 x \cos (x) + 2 \sin (x)$ with respect to $x$ is $\frac{d}{d x} [- x^{2} \sin (x)] + \frac{d}{d x} [4 x \cos (x)] + \frac{d}{d x} [2 \sin (x)]$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.3

Evaluate $\frac{d}{d x} [4 x \cos (x)]$ .

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Step 3.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x \cos (x)$ with respect to $x$ is $4 \frac{d}{d x} [x \cos (x)]$ .

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

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

$f''' (x) = - (x^{2} \cos (x) + \sin (x) (2 x)) + 4 (x (- \sin (x)) + \cos (x) \cdot 1) + \frac{d}{d x} (2 \sin (x))$

Step 3.3.5

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

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

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

Step 3.5.2

Apply the distributive property.

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

Step 3.5.3

Combine terms.

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Step 3.5.3.1

Multiply $2$ by $- 1$ .

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

Step 3.5.3.2

Multiply $- 1$ by $4$ .

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

Step 3.5.3.3

Subtract $4 x \sin (x)$ from $- 2 \sin (x) x$ .

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Step 3.5.3.3.1

Move $\sin (x)$ .

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

Step 3.5.3.3.2

Subtract $4 x \sin (x)$ from $- 2 x \sin (x)$ .

$f''' (x) = - x^{2} \cos (x) - 6 x \sin (x) + 4 \cos (x) + 2 \cos (x)$

Step 3.5.3.4

Add $4 \cos (x)$ and $2 \cos (x)$ .

$f''' (x) = - x^{2} \cos (x) - 6 x \sin (x) + 6 \cos (x)$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $- x^{2} \cos (x) - 6 x \sin (x) + 6 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [- x^{2} \cos (x)] + \frac{d}{d x} [- 6 x \sin (x)] + \frac{d}{d x} [6 \cos (x)]$ .

$f^{4} (x) = \frac{d}{d x} (- x^{2} \cos (x)) + \frac{d}{d x} (- 6 x \sin (x)) + \frac{d}{d x} (6 \cos (x))$

Step 4.2

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

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Step 4.2.1

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

$f^{4} (x) = - \frac{d}{d x} (x^{2} \cos (x)) + \frac{d}{d x} (- 6 x \sin (x)) + \frac{d}{d x} (6 \cos (x))$

Step 4.2.2

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

$f^{4} (x) = - (x^{2} \frac{d}{d x} (\cos (x)) + \cos (x) \frac{d}{d x} (x^{2})) + \frac{d}{d x} (- 6 x \sin (x)) + \frac{d}{d x} (6 \cos (x))$

Step 4.2.3

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

$f^{4} (x) = - (x^{2} (- \sin (x)) + \cos (x) \frac{d}{d x} (x^{2})) + \frac{d}{d x} (- 6 x \sin (x)) + \frac{d}{d x} (6 \cos (x))$

Step 4.2.4

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

$f^{4} (x) = - (x^{2} (- \sin (x)) + \cos (x) (2 x)) + \frac{d}{d x} (- 6 x \sin (x)) + \frac{d}{d x} (6 \cos (x))$

Step 4.3

Evaluate $\frac{d}{d x} [- 6 x \sin (x)]$ .

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Step 4.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x \sin (x)$ with respect to $x$ is $- 6 \frac{d}{d x} [x \sin (x)]$ .

$f^{4} (x) = - (x^{2} (- \sin (x)) + \cos (x) (2 x)) - 6 \frac{d}{d x} (x \sin (x)) + \frac{d}{d x} (6 \cos (x))$

Step 4.3.2

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

$f^{4} (x) = - (x^{2} (- \sin (x)) + \cos (x) (2 x)) - 6 (x \frac{d}{d x} (\sin (x)) + \sin (x) \frac{d}{d x} (x)) + \frac{d}{d x} (6 \cos (x))$

Step 4.3.3

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

$f^{4} (x) = - (x^{2} (- \sin (x)) + \cos (x) (2 x)) - 6 (x \cos (x) + \sin (x) \frac{d}{d x} (x)) + \frac{d}{d x} (6 \cos (x))$

Step 4.3.4

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

$f^{4} (x) = - (x^{2} (- \sin (x)) + \cos (x) (2 x)) - 6 (x \cos (x) + \sin (x) \cdot 1) + \frac{d}{d x} (6 \cos (x))$

Step 4.3.5

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

$f^{4} (x) = - (x^{2} (- \sin (x)) + \cos (x) (2 x)) - 6 (x \cos (x) + \sin (x)) + \frac{d}{d x} (6 \cos (x))$

Step 4.4

Evaluate $\frac{d}{d x} [6 \cos (x)]$ .

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Step 4.4.1

Since $6$ is constant with respect to $x$ , the derivative of $6 \cos (x)$ with respect to $x$ is $6 \frac{d}{d x} [\cos (x)]$ .

$f^{4} (x) = - (x^{2} (- \sin (x)) + \cos (x) (2 x)) - 6 (x \cos (x) + \sin (x)) + 6 \frac{d}{d x} (\cos (x))$

Step 4.4.2

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

$f^{4} (x) = - (x^{2} (- \sin (x)) + \cos (x) (2 x)) - 6 (x \cos (x) + \sin (x)) + 6 (- \sin (x))$

Step 4.4.3

Multiply $- 1$ by $6$ .

$f^{4} (x) = - (x^{2} (- \sin (x)) + \cos (x) (2 x)) - 6 (x \cos (x) + \sin (x)) - 6 \sin (x)$

Step 4.5

Simplify.

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Step 4.5.1

Apply the distributive property.

$f^{4} (x) = - (x^{2} (- \sin (x))) - (\cos (x) (2 x)) - 6 (x \cos (x) + \sin (x)) - 6 \sin (x)$

Step 4.5.2

Apply the distributive property.

$f^{4} (x) = - (x^{2} (- \sin (x))) - (\cos (x) (2 x)) - 6 (x \cos (x)) - 6 \sin (x) - 6 \sin (x)$

Step 4.5.3

Combine terms.

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Step 4.5.3.1

Multiply $- 1$ by $- 1$ .

$f^{4} (x) = 1 (x^{2} (\sin (x))) - (\cos (x) (2 x)) - 6 (x \cos (x)) - 6 \sin (x) - 6 \sin (x)$

Step 4.5.3.2

Multiply $x^{2}$ by $1$ .

$f^{4} (x) = x^{2} \sin (x) - (\cos (x) (2 x)) - 6 (x \cos (x)) - 6 \sin (x) - 6 \sin (x)$

Step 4.5.3.3

Multiply $2$ by $- 1$ .

$f^{4} (x) = x^{2} \sin (x) - 2 (\cos (x) (x)) - 6 (x \cos (x)) - 6 \sin (x) - 6 \sin (x)$

Step 4.5.3.4

Subtract $6 x \cos (x)$ from $- 2 \cos (x) x$ .

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Step 4.5.3.4.1

Move $\cos (x)$ .

$f^{4} (x) = x^{2} \sin (x) - 2 x \cos (x) - 6 x \cos (x) - 6 \sin (x) - 6 \sin (x)$

Step 4.5.3.4.2

Subtract $6 x \cos (x)$ from $- 2 x \cos (x)$ .

$f^{4} (x) = x^{2} \sin (x) - 8 x \cos (x) - 6 \sin (x) - 6 \sin (x)$

Step 4.5.3.5

Subtract $6 \sin (x)$ from $- 6 \sin (x)$ .

$f^{4} (x) = x^{2} \sin (x) - 8 x \cos (x) - 12 \sin (x)$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $x^{2} \sin (x) - 8 x \cos (x) - 12 \sin (x)$ .

$x^{2} \sin (x) - 8 x \cos (x) - 12 \sin (x)$