Calculus Examples

$y = 2 \sin (x) \cos (x)$ ; $x = \frac{π}{2}$

Step 1

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

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

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Add $1$ and $1$ .

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Add $1$ and $1$ .

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

Step 13

Simplify.

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Step 13.1

Apply the distributive property.

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

Step 13.2

Multiply $- 1$ by $2$ .

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

Step 14

Evaluate the derivative at $x = \frac{π}{2}$ .

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

Step 15

Simplify.

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Step 15.1

Simplify each term.

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Step 15.1.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

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

Step 15.1.2

One to any power is one.

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

Step 15.1.3

Multiply $- 2$ by $1$ .

$- 2 + 2 \cos^{2} (\frac{π}{2})$

Step 15.1.4

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$- 2 + 2 \cdot 0^{2}$

Step 15.1.5

Raising $0$ to any positive power yields $0$ .

$- 2 + 2 \cdot 0$

Step 15.1.6

Multiply $2$ by $0$ .

$- 2 + 0$

Step 15.2

Add $- 2$ and $0$ .

$- 2$