Calculus Examples

$y = 4 \sin (x) \cos (x)$ ; $x = \frac{π}{4}$

Step 1

Find the corresponding $y$ -value to $x = \frac{π}{4}$ .

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Step 1.1

Substitute $\frac{π}{4}$ in for $x$ .

$y = 4 \sin (\frac{π}{4}) \cos (\frac{π}{4})$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 4 \sin (\frac{π}{4}) \cos (\frac{π}{4})$

Step 1.2.2

Simplify $4 \sin (\frac{π}{4}) \cos (\frac{π}{4})$ .

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Step 1.2.2.1

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$y = 4 \frac{\sqrt{2}}{2} \cos (\frac{π}{4})$

Step 1.2.2.2

Cancel the common factor of $2$ .

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Step 1.2.2.2.1

Factor $2$ out of $4$ .

$y = 2 (2) \frac{\sqrt{2}}{2} \cos (\frac{π}{4})$

Step 1.2.2.2.2

Cancel the common factor.

$y = 2 \cdot 2 \frac{\sqrt{2}}{2} \cos (\frac{π}{4})$

Step 1.2.2.2.3

Rewrite the expression.

$y = 2 \sqrt{2} \cos (\frac{π}{4})$

Step 1.2.2.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$y = 2 \sqrt{2} \frac{\sqrt{2}}{2}$

Step 1.2.2.4

Cancel the common factor of $2$ .

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Step 1.2.2.4.1

Factor $2$ out of $2 \sqrt{2}$ .

$y = 2 (\sqrt{2}) \frac{\sqrt{2}}{2}$

Step 1.2.2.4.2

Cancel the common factor.

$y = 2 \sqrt{2} \frac{\sqrt{2}}{2}$

Step 1.2.2.4.3

Rewrite the expression.

$y = \sqrt{2} \sqrt{2}$

Step 1.2.2.5

Raise $\sqrt{2}$ to the power of $1$ .

$y = {\sqrt{2}}^{1} \sqrt{2}$

Step 1.2.2.6

Raise $\sqrt{2}$ to the power of $1$ .

$y = {\sqrt{2}}^{1} {\sqrt{2}}^{1}$

Step 1.2.2.7

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

$y = {\sqrt{2}}^{1 + 1}$

Step 1.2.2.8

Add $1$ and $1$ .

$y = {\sqrt{2}}^{2}$

Step 1.2.2.9

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

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Step 1.2.2.9.1

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

$y = {(2^{\frac{1}{2}})}^{2}$

Step 1.2.2.9.2

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

$y = 2^{\frac{1}{2} \cdot 2}$

Step 1.2.2.9.3

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

$y = 2^{\frac{2}{2}}$

Step 1.2.2.9.4

Cancel the common factor of $2$ .

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Step 1.2.2.9.4.1

Cancel the common factor.

$y = 2^{\frac{2}{2}}$

Step 1.2.2.9.4.2

Rewrite the expression.

$y = 2^{1}$

Step 1.2.2.9.5

Evaluate the exponent.

$y = 2$

Step 2

Find the first derivative and evaluate at $x = \frac{π}{4}$ and $y = 2$ to find the slope of the tangent line.

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Step 2.1

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

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

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

$4 (\sin (x) \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 2.3

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

$4 (\sin (x) (- \sin (x)) + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 2.4

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

$4 (- (\sin^{1} (x) \sin (x)) + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 2.5

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

$4 (- (\sin^{1} (x) \sin^{1} (x)) + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 2.6

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

$4 (- {\sin (x)}^{1 + 1} + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 2.7

Add $1$ and $1$ .

$4 (- \sin^{2} (x) + \cos (x) \frac{d}{d x} [\sin (x)])$

Step 2.8

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

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

Step 2.9

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

$4 (- \sin^{2} (x) + \cos^{1} (x) \cos (x))$

Step 2.10

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

$4 (- \sin^{2} (x) + \cos^{1} (x) \cos^{1} (x))$

Step 2.11

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

$4 (- \sin^{2} (x) + {\cos (x)}^{1 + 1})$

Step 2.12

Add $1$ and $1$ .

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

Step 2.13

Simplify.

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Step 2.13.1

Apply the distributive property.

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

Step 2.13.2

Multiply $- 1$ by $4$ .

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

Step 2.13.3

Rewrite $4 \cos^{2} (x)$ as ${(2 \cos (x))}^{2}$ .

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

Step 2.13.4

Rewrite $4 \sin^{2} (x)$ as ${(2 \sin (x))}^{2}$ .

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

Step 2.13.5

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

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

Step 2.13.6

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

$(2 \cos (x) + 2 \sin (x)) (2 \cos (x) - (2 \sin (x)))$

Step 2.13.7

Multiply $2$ by $- 1$ .

$(2 \cos (x) + 2 \sin (x)) (2 \cos (x) - 2 \sin (x))$

Step 2.13.8

Expand $(2 \cos (x) + 2 \sin (x)) (2 \cos (x) - 2 \sin (x))$ using the FOIL Method.

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Step 2.13.8.1

Apply the distributive property.

$2 \cos (x) (2 \cos (x) - 2 \sin (x)) + 2 \sin (x) (2 \cos (x) - 2 \sin (x))$

Step 2.13.8.2

Apply the distributive property.

$2 \cos (x) (2 \cos (x)) + 2 \cos (x) (- 2 \sin (x)) + 2 \sin (x) (2 \cos (x) - 2 \sin (x))$

Step 2.13.8.3

Apply the distributive property.

$2 \cos (x) (2 \cos (x)) + 2 \cos (x) (- 2 \sin (x)) + 2 \sin (x) (2 \cos (x)) + 2 \sin (x) (- 2 \sin (x))$

Step 2.13.9

Combine the opposite terms in $2 \cos (x) (2 \cos (x)) + 2 \cos (x) (- 2 \sin (x)) + 2 \sin (x) (2 \cos (x)) + 2 \sin (x) (- 2 \sin (x))$ .

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Step 2.13.9.1

Reorder the factors in the terms $2 \cos (x) (- 2 \sin (x))$ and $2 \sin (x) (2 \cos (x))$ .

$2 \cos (x) (2 \cos (x)) - 2 \cdot 2 \cos (x) \sin (x) + 2 \cdot 2 \cos (x) \sin (x) + 2 \sin (x) (- 2 \sin (x))$

Step 2.13.9.2

Add $- 2 \cdot 2 \cos (x) \sin (x)$ and $2 \cdot 2 \cos (x) \sin (x)$ .

$2 \cos (x) (2 \cos (x)) + 0 + 2 \sin (x) (- 2 \sin (x))$

Step 2.13.9.3

Add $2 \cos (x) (2 \cos (x))$ and $0$ .

$2 \cos (x) (2 \cos (x)) + 2 \sin (x) (- 2 \sin (x))$

Step 2.13.10

Simplify each term.

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Step 2.13.10.1

Multiply $2 \cos (x) (2 \cos (x))$ .

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Step 2.13.10.1.1

Multiply $2$ by $2$ .

$4 \cos (x) \cos (x) + 2 \sin (x) (- 2 \sin (x))$

Step 2.13.10.1.2

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

$4 (\cos^{1} (x) \cos (x)) + 2 \sin (x) (- 2 \sin (x))$

Step 2.13.10.1.3

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

$4 (\cos^{1} (x) \cos^{1} (x)) + 2 \sin (x) (- 2 \sin (x))$

Step 2.13.10.1.4

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

$4 {\cos (x)}^{1 + 1} + 2 \sin (x) (- 2 \sin (x))$

Step 2.13.10.1.5

Add $1$ and $1$ .

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

Step 2.13.10.2

Multiply $2 \sin (x) (- 2 \sin (x))$ .

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Step 2.13.10.2.1

Multiply $- 2$ by $2$ .

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

Step 2.13.10.2.2

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

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

Step 2.13.10.2.3

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

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

Step 2.13.10.2.4

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

$4 \cos^{2} (x) - 4 {\sin (x)}^{1 + 1}$

Step 2.13.10.2.5

Add $1$ and $1$ .

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

Step 2.14

Evaluate the derivative at $x = \frac{π}{4}$ .

$4 \cos^{2} (\frac{π}{4}) - 4 \sin^{2} (\frac{π}{4})$

Step 2.15

Simplify.

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Step 2.15.1

Simplify each term.

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Step 2.15.1.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$4 {(\frac{\sqrt{2}}{2})}^{2} - 4 \sin^{2} (\frac{π}{4})$

Step 2.15.1.2

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

$4 \frac{{\sqrt{2}}^{2}}{2^{2}} - 4 \sin^{2} (\frac{π}{4})$

Step 2.15.1.3

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

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Step 2.15.1.3.1

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

$4 \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}} - 4 \sin^{2} (\frac{π}{4})$

Step 2.15.1.3.2

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

$4 \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}} - 4 \sin^{2} (\frac{π}{4})$

Step 2.15.1.3.3

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

$4 \frac{2^{\frac{2}{2}}}{2^{2}} - 4 \sin^{2} (\frac{π}{4})$

Step 2.15.1.3.4

Cancel the common factor of $2$ .

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Step 2.15.1.3.4.1

Cancel the common factor.

$4 \frac{2^{\frac{2}{2}}}{2^{2}} - 4 \sin^{2} (\frac{π}{4})$

Step 2.15.1.3.4.2

Rewrite the expression.

$4 \frac{2^{1}}{2^{2}} - 4 \sin^{2} (\frac{π}{4})$

Step 2.15.1.3.5

Evaluate the exponent.

$4 \frac{2}{2^{2}} - 4 \sin^{2} (\frac{π}{4})$

Step 2.15.1.4

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

$4 (\frac{2}{4}) - 4 \sin^{2} (\frac{π}{4})$

Step 2.15.1.5

Cancel the common factor of $4$ .

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Step 2.15.1.5.1

Cancel the common factor.

$4 (\frac{2}{4}) - 4 \sin^{2} (\frac{π}{4})$

Step 2.15.1.5.2

Rewrite the expression.

$2 - 4 \sin^{2} (\frac{π}{4})$

Step 2.15.1.6

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$2 - 4 {(\frac{\sqrt{2}}{2})}^{2}$

Step 2.15.1.7

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

$2 - 4 \frac{{\sqrt{2}}^{2}}{2^{2}}$

Step 2.15.1.8

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

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Step 2.15.1.8.1

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

$2 - 4 \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 2.15.1.8.2

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

$2 - 4 \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 2.15.1.8.3

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

$2 - 4 \frac{2^{\frac{2}{2}}}{2^{2}}$

Step 2.15.1.8.4

Cancel the common factor of $2$ .

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Step 2.15.1.8.4.1

Cancel the common factor.

$2 - 4 \frac{2^{\frac{2}{2}}}{2^{2}}$

Step 2.15.1.8.4.2

Rewrite the expression.

$2 - 4 \frac{2^{1}}{2^{2}}$

Step 2.15.1.8.5

Evaluate the exponent.

$2 - 4 \frac{2}{2^{2}}$

Step 2.15.1.9

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

$2 - 4 (\frac{2}{4})$

Step 2.15.1.10

Cancel the common factor of $4$ .

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Step 2.15.1.10.1

Factor $4$ out of $- 4$ .

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

Step 2.15.1.10.2

Cancel the common factor.

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

Step 2.15.1.10.3

Rewrite the expression.

$2 - 1 \cdot 2$

Step 2.15.1.11

Multiply $- 1$ by $2$ .

$2 - 2$

Step 2.15.2

Subtract $2$ from $2$ .

$0$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $0$ and a given point $(\frac{π}{4}, 2)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = 0 \cdot (x - (\frac{π}{4}))$

Step 3.2

Simplify the equation and keep it in point-slope form.

$y - 2 = 0 \cdot (x - \frac{π}{4})$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Multiply $0$ by $x - \frac{π}{4}$ .

$y - 2 = 0$

Step 3.3.2

Add $2$ to both sides of the equation.

$y = 2$

Step 4