Calculus Examples

$y = 2 \sin (x) \cos (x)$ ; $x = \frac{π}{3}$

Step 1

Find the corresponding $y$ -value to $x = \frac{π}{3}$ .

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Step 1.1

Substitute $\frac{π}{3}$ in for $x$ .

$y = 2 \sin (\frac{π}{3}) \cos (\frac{π}{3})$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 2 \sin (\frac{π}{3}) \cos (\frac{π}{3})$

Step 1.2.2

Simplify $2 \sin (\frac{π}{3}) \cos (\frac{π}{3})$ .

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Step 1.2.2.1

Apply the sine double-angle identity.

$y = \sin (2 \frac{π}{3})$

Step 1.2.2.2

Combine $2$ and $\frac{π}{3}$ .

$y = \sin (\frac{2 π}{3})$

Step 1.2.2.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$y = \sin (\frac{π}{3})$

Step 1.2.2.4

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

$y = \frac{\sqrt{3}}{2}$

Step 2

Find the first derivative and evaluate at $x = \frac{π}{3}$ and $y = \frac{\sqrt{3}}{2}$ to find the slope of the tangent line.

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Step 2.1

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 2.2

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 2.3

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 2.4

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

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

Step 2.5

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

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

Step 2.6

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 2.7

Add $1$ and $1$ .

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

Add $1$ and $1$ .

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

Step 2.13

Simplify.

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Step 2.13.1

Apply the distributive property.

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

Step 2.13.2

Multiply $- 1$ by $2$ .

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

Step 2.14

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

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

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

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

Step 2.15.1.2

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

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

Step 2.15.1.3

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

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Step 2.15.1.3.1

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

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

Step 2.15.1.3.2

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

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

Step 2.15.1.3.3

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

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

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.

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

Step 2.15.1.3.4.2

Rewrite the expression.

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

Step 2.15.1.3.5

Evaluate the exponent.

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

Step 2.15.1.4

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

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

Step 2.15.1.5

Cancel the common factor of $2$ .

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Step 2.15.1.5.1

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{3}{4} + 2 \cos^{2} (\frac{π}{3})$

Step 2.15.1.5.2

Factor $2$ out of $4$ .

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

Step 2.15.1.5.3

Cancel the common factor.

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

Step 2.15.1.5.4

Rewrite the expression.

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

Step 2.15.1.6

Rewrite $- 1 (\frac{3}{2})$ as $- (\frac{3}{2})$ .

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

Step 2.15.1.7

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

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

Step 2.15.1.8

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

$- \frac{3}{2} + 2 \frac{1^{2}}{2^{2}}$

Step 2.15.1.9

One to any power is one.

$- \frac{3}{2} + 2 \frac{1}{2^{2}}$

Step 2.15.1.10

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

$- \frac{3}{2} + 2 (\frac{1}{4})$

Step 2.15.1.11

Cancel the common factor of $2$ .

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Step 2.15.1.11.1

Factor $2$ out of $4$ .

$- \frac{3}{2} + 2 \frac{1}{2 (2)}$

Step 2.15.1.11.2

Cancel the common factor.

$- \frac{3}{2} + 2 \frac{1}{2 \cdot 2}$

Step 2.15.1.11.3

Rewrite the expression.

$- \frac{3}{2} + \frac{1}{2}$

Step 2.15.2

Combine fractions.

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Step 2.15.2.1

Combine the numerators over the common denominator.

$\frac{- 3 + 1}{2}$

Step 2.15.2.2

Simplify the expression.

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Step 2.15.2.2.1

Add $- 3$ and $1$ .

$\frac{- 2}{2}$

Step 2.15.2.2.2

Divide $- 2$ by $2$ .

$- 1$

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 $- 1$ and a given point $(\frac{π}{3}, \frac{\sqrt{3}}{2})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{\sqrt{3}}{2}) = - 1 \cdot (x - (\frac{π}{3}))$

Step 3.2

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

$y - \frac{\sqrt{3}}{2} = - 1 \cdot (x - \frac{π}{3})$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- 1 \cdot (x - \frac{π}{3})$ .

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Step 3.3.1.1

Rewrite.

$y - \frac{\sqrt{3}}{2} = 0 + 0 - 1 \cdot (x - \frac{π}{3})$

Step 3.3.1.2

Simplify by adding zeros.

$y - \frac{\sqrt{3}}{2} = - 1 \cdot (x - \frac{π}{3})$

Step 3.3.1.3

Apply the distributive property.

$y - \frac{\sqrt{3}}{2} = - 1 x - 1 (- \frac{π}{3})$

Step 3.3.1.4

Rewrite $- 1 x$ as $- x$ .

$y - \frac{\sqrt{3}}{2} = - x - 1 (- \frac{π}{3})$

Step 3.3.1.5

Multiply $- 1 (- \frac{π}{3})$ .

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Step 3.3.1.5.1

Multiply $- 1$ by $- 1$ .

$y - \frac{\sqrt{3}}{2} = - x + 1 \frac{π}{3}$

Step 3.3.1.5.2

Multiply $\frac{π}{3}$ by $1$ .

$y - \frac{\sqrt{3}}{2} = - x + \frac{π}{3}$

Step 3.3.2

Add $\frac{\sqrt{3}}{2}$ to both sides of the equation.

$y = - x + \frac{π}{3} + \frac{\sqrt{3}}{2}$

Step 3.3.3

Write in $y = m x + b$ form.

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Step 3.3.3.1

To write $\frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - x + \frac{π}{3} \cdot \frac{2}{2} + \frac{\sqrt{3}}{2}$

Step 3.3.3.2

To write $\frac{\sqrt{3}}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - x + \frac{π}{3} \cdot \frac{2}{2} + \frac{\sqrt{3}}{2} \cdot \frac{3}{3}$

Step 3.3.3.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.3.3.1

Multiply $\frac{π}{3}$ by $\frac{2}{2}$ .

$y = - x + \frac{π \cdot 2}{3 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{3}{3}$

Step 3.3.3.3.2

Multiply $3$ by $2$ .

$y = - x + \frac{π \cdot 2}{6} + \frac{\sqrt{3}}{2} \cdot \frac{3}{3}$

Step 3.3.3.3.3

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{3}{3}$ .

$y = - x + \frac{π \cdot 2}{6} + \frac{\sqrt{3} \cdot 3}{2 \cdot 3}$

Step 3.3.3.3.4

Multiply $2$ by $3$ .

$y = - x + \frac{π \cdot 2}{6} + \frac{\sqrt{3} \cdot 3}{6}$

Step 3.3.3.4

Combine the numerators over the common denominator.

$y = - x + \frac{π \cdot 2 + \sqrt{3} \cdot 3}{6}$

Step 3.3.3.5

Simplify the numerator.

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Step 3.3.3.5.1

Move $2$ to the left of $π$ .

$y = - x + \frac{2 \cdot π + \sqrt{3} \cdot 3}{6}$

Step 3.3.3.5.2

Move $3$ to the left of $\sqrt{3}$ .

$y = - x + \frac{2 π + 3 \sqrt{3}}{6}$

Step 4