Calculus Examples

$\sin (y) = 5 x^{4} - 5$ , $(1, π)$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = π$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (\sin (y)) = \frac{d}{d x} (5 x^{4} - 5)$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

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

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Step 1.2.1.1

To apply the Chain Rule, set $u$ as $y$ .

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

Step 1.2.1.2

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

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

Step 1.2.1.3

Replace all occurrences of $u$ with $y$ .

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

Step 1.2.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$\cos (y) y'$

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

By the Sum Rule, the derivative of $5 x^{4} - 5$ with respect to $x$ is $\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [- 5]$ .

$\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [- 5]$

Step 1.3.2

Evaluate $\frac{d}{d x} [5 x^{4}]$ .

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Step 1.3.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{4}$ with respect to $x$ is $5 \frac{d}{d x} [x^{4}]$ .

$5 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 5]$

Step 1.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$5 (4 x^{3}) + \frac{d}{d x} [- 5]$

Step 1.3.2.3

Multiply $4$ by $5$ .

$20 x^{3} + \frac{d}{d x} [- 5]$

Step 1.3.3

Differentiate using the Constant Rule.

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Step 1.3.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$20 x^{3} + 0$

Step 1.3.3.2

Add $20 x^{3}$ and $0$ .

$20 x^{3}$

Step 1.4

Reform the equation by setting the left side equal to the right side.

$\cos (y) y' = 20 x^{3}$

Step 1.5

Divide each term in $\cos (y) y' = 20 x^{3}$ by $\cos (y)$ and simplify.

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Step 1.5.1

Divide each term in $\cos (y) y' = 20 x^{3}$ by $\cos (y)$ .

$\frac{\cos (y) y'}{\cos (y)} = \frac{20 x^{3}}{\cos (y)}$

Step 1.5.2

Simplify the left side.

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Step 1.5.2.1

Cancel the common factor of $\cos (y)$ .

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Step 1.5.2.1.1

Cancel the common factor.

$\frac{\cos (y) y'}{\cos (y)} = \frac{20 x^{3}}{\cos (y)}$

Step 1.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{20 x^{3}}{\cos (y)}$

Step 1.6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{20 x^{3}}{\cos (y)}$

Step 1.7

Evaluate at $x = 1$ and $y = π$ .

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Step 1.7.1

Replace the variable $x$ with $1$ in the expression.

$\frac{20 {(1)}^{3}}{\cos (y)}$

Step 1.7.2

Replace the variable $y$ with $π$ in the expression.

$\frac{20 {(1)}^{3}}{\cos (π)}$

Step 1.7.3

One to any power is one.

$\frac{20 \cdot 1}{\cos (π)}$

Step 1.7.4

Simplify the denominator.

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Step 1.7.4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{20 \cdot 1}{- \cos (0)}$

Step 1.7.4.2

The exact value of $\cos (0)$ is $1$ .

$\frac{20 \cdot 1}{- 1 \cdot 1}$

Step 1.7.4.3

Multiply $- 1$ by $1$ .

$\frac{20 \cdot 1}{- 1}$

Step 1.7.5

Simplify the expression.

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Step 1.7.5.1

Multiply $20$ by $1$ .

$\frac{20}{- 1}$

Step 1.7.5.2

Divide $20$ by $- 1$ .

$- 20$

Step 2

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

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Step 2.1

Use the slope $- 20$ and a given point $(1, π)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (π) = - 20 \cdot (x - (1))$

Step 2.2

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

$y - π = - 20 \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 20 \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y - π = 0 + 0 - 20 \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - π = - 20 \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y - π = - 20 x - 20 \cdot - 1$

Step 2.3.1.4

Multiply $- 20$ by $- 1$ .

$y - π = - 20 x + 20$

Step 2.3.2

Add $π$ to both sides of the equation.

$y = - 20 x + 20 + π$

Step 3