Calculus Examples

$y = \sin (x) \cos (x)$

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 $f (x) = \sin (x)$ and $g (x) = \cos (x)$ .

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

Step 1.2

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

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

Step 1.3

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.4

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.5

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

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

Step 1.6

Add $1$ and $1$ .

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

Step 1.7

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

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

Step 1.8

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.9

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.10

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

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

Step 1.11

Add $1$ and $1$ .

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

Step 1.12

Simplify.

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Step 1.12.1

Reorder $- \sin^{2} (x)$ and $\cos^{2} (x)$ .

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

Step 1.12.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (x)$ and $b = \sin (x)$ .

$(\cos (x) + \sin (x)) (\cos (x) - \sin (x))$

Step 1.12.3

Expand $(\cos (x) + \sin (x)) (\cos (x) - \sin (x))$ using the FOIL Method.

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Step 1.12.3.1

Apply the distributive property.

$\cos (x) (\cos (x) - \sin (x)) + \sin (x) (\cos (x) - \sin (x))$

Step 1.12.3.2

Apply the distributive property.

$\cos (x) \cos (x) + \cos (x) (- \sin (x)) + \sin (x) (\cos (x) - \sin (x))$

Step 1.12.3.3

Apply the distributive property.

$\cos (x) \cos (x) + \cos (x) (- \sin (x)) + \sin (x) \cos (x) + \sin (x) (- \sin (x))$

Step 1.12.4

Combine the opposite terms in $\cos (x) \cos (x) + \cos (x) (- \sin (x)) + \sin (x) \cos (x) + \sin (x) (- \sin (x))$ .

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Step 1.12.4.1

Reorder the factors in the terms $\cos (x) (- \sin (x))$ and $\sin (x) \cos (x)$ .

$\cos (x) \cos (x) - \cos (x) \sin (x) + \cos (x) \sin (x) + \sin (x) (- \sin (x))$

Step 1.12.4.2

Add $- \cos (x) \sin (x)$ and $\cos (x) \sin (x)$ .

$\cos (x) \cos (x) + 0 + \sin (x) (- \sin (x))$

Step 1.12.4.3

Add $\cos (x) \cos (x)$ and $0$ .

$\cos (x) \cos (x) + \sin (x) (- \sin (x))$

Step 1.12.5

Simplify each term.

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Step 1.12.5.1

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

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Step 1.12.5.1.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.12.5.1.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.12.5.1.3

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

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

Step 1.12.5.1.4

Add $1$ and $1$ .

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

Step 1.12.5.2

Rewrite using the commutative property of multiplication.

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

Step 1.12.5.3

Multiply $- \sin (x) \sin (x)$ .

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Step 1.12.5.3.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.12.5.3.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.12.5.3.3

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

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

Step 1.12.5.3.4

Add $1$ and $1$ .

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

Step 1.12.6

Apply the cosine double-angle identity.

$f' (x) = \cos (2 x)$

Step 2

Find the second derivative.

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Step 2.1

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

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Step 2.1.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 2.1.2

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

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

Step 2.1.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2.3

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

$- 2 \sin (2 x) \cdot 1$

Step 2.2.4

Multiply $- 2$ by $1$ .

$f'' (x) = - 2 \sin (2 x)$