Calculus Examples

$y = 8 \sin (x) \cos (x)$ , $x = \frac{π}{3}$

Step 1

Since $8$ is constant with respect to $x$ , the derivative of $8 \sin (x) \cos (x)$ with respect to $x$ is $8 \frac{d}{d x} [\sin (x) \cos (x)]$ .

$8 \frac{d}{d x} [\sin (x) \cos (x)]$

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

$8 (\sin (x) \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 3

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

$8 (\sin (x) (- \sin (x)) + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 4

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

$8 (- (\sin^{1} (x) \sin (x)) + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 5

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

$8 (- (\sin^{1} (x) \sin^{1} (x)) + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 6

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

$8 (- {\sin (x)}^{1 + 1} + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 7

Add $1$ and $1$ .

$8 (- \sin^{2} (x) + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 8

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

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

Step 9

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

$8 (- \sin^{2} (x) + \cos^{1} (x) \cos (x))$

Step 10

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

$8 (- \sin^{2} (x) + \cos^{1} (x) \cos^{1} (x))$

Step 11

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

$8 (- \sin^{2} (x) + {\cos (x)}^{1 + 1})$

Step 12

Add $1$ and $1$ .

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

Step 13

Simplify.

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Step 13.1

Apply the distributive property.

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

Step 13.2

Multiply $- 1$ by $8$ .

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

Step 14

Evaluate the derivative at $x = \frac{π}{3}$ .

$- 8 \sin^{2} (\frac{π}{3}) + 8 \cos^{2} (\frac{π}{3})$

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 $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$- 8 {(\frac{\sqrt{3}}{2})}^{2} + 8 \cos^{2} (\frac{π}{3})$

Step 15.1.2

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

$- 8 \frac{{\sqrt{3}}^{2}}{2^{2}} + 8 \cos^{2} (\frac{π}{3})$

Step 15.1.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 15.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- 8 \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}} + 8 \cos^{2} (\frac{π}{3})$

Step 15.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 8 \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}} + 8 \cos^{2} (\frac{π}{3})$

Step 15.1.3.3

Combine $\frac{1}{2}$ and $2$ .

$- 8 \frac{3^{\frac{2}{2}}}{2^{2}} + 8 \cos^{2} (\frac{π}{3})$

Step 15.1.3.4

Cancel the common factor of $2$ .

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Step 15.1.3.4.1

Cancel the common factor.

$- 8 \frac{3^{\frac{2}{2}}}{2^{2}} + 8 \cos^{2} (\frac{π}{3})$

Step 15.1.3.4.2

Rewrite the expression.

$- 8 \frac{3^{1}}{2^{2}} + 8 \cos^{2} (\frac{π}{3})$

Step 15.1.3.5

Evaluate the exponent.

$- 8 \frac{3}{2^{2}} + 8 \cos^{2} (\frac{π}{3})$

Step 15.1.4

Raise $2$ to the power of $2$ .

$- 8 (\frac{3}{4}) + 8 \cos^{2} (\frac{π}{3})$

Step 15.1.5

Cancel the common factor of $4$ .

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Step 15.1.5.1

Factor $4$ out of $- 8$ .

$4 (- 2) \frac{3}{4} + 8 \cos^{2} (\frac{π}{3})$

Step 15.1.5.2

Cancel the common factor.

$4 \cdot - 2 \frac{3}{4} + 8 \cos^{2} (\frac{π}{3})$

Step 15.1.5.3

Rewrite the expression.

$- 2 \cdot 3 + 8 \cos^{2} (\frac{π}{3})$

Step 15.1.6

Multiply $- 2$ by $3$ .

$- 6 + 8 \cos^{2} (\frac{π}{3})$

Step 15.1.7

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$- 6 + 8 {(\frac{1}{2})}^{2}$

Step 15.1.8

Apply the product rule to $\frac{1}{2}$ .

$- 6 + 8 \frac{1^{2}}{2^{2}}$

Step 15.1.9

One to any power is one.

$- 6 + 8 \frac{1}{2^{2}}$

Step 15.1.10

Raise $2$ to the power of $2$ .

$- 6 + 8 (\frac{1}{4})$

Step 15.1.11

Cancel the common factor of $4$ .

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Step 15.1.11.1

Factor $4$ out of $8$ .

$- 6 + 4 (2) \frac{1}{4}$

Step 15.1.11.2

Cancel the common factor.

$- 6 + 4 \cdot 2 \frac{1}{4}$

Step 15.1.11.3

Rewrite the expression.

$- 6 + 2$

Step 15.2

Add $- 6$ and $2$ .

$- 4$