Calculus Examples

$y = 3 \sin (π x + y)$ , $(1, 0)$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = 0$ to find the slope of the tangent line.

Tap for more steps...

Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (3 \sin (π x + y))$

Step 1.2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 1.3

Differentiate the right side of the equation.

Tap for more steps...

Step 1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 \sin (π x + y)$ with respect to $x$ is $3 \frac{d}{d x} [\sin (π x + y)]$ .

$3 \frac{d}{d x} [\sin (π x + y)]$

Step 1.3.2

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 $g (x) = π x + y$ .

Tap for more steps...

Step 1.3.2.1

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

$3 (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [π x + y])$

Step 1.3.2.2

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

$3 (\cos (u) \frac{d}{d x} [π x + y])$

Step 1.3.2.3

Replace all occurrences of $u$ with $π x + y$ .

$3 (\cos (π x + y) \frac{d}{d x} [π x + y])$

Step 1.3.3

Differentiate.

Tap for more steps...

Step 1.3.3.1

By the Sum Rule, the derivative of $π x + y$ with respect to $x$ is $\frac{d}{d x} [π x] + \frac{d}{d x} [y]$ .

$3 \cos (π x + y) (\frac{d}{d x} [π x] + \frac{d}{d x} [y])$

Step 1.3.3.2

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

$3 \cos (π x + y) (π \frac{d}{d x} [x] + \frac{d}{d x} [y])$

Step 1.3.3.3

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

$3 \cos (π x + y) (π \cdot 1 + \frac{d}{d x} [y])$

Step 1.3.3.4

Multiply $π$ by $1$ .

$3 \cos (π x + y) (π + \frac{d}{d x} [y])$

Step 1.3.4

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

$3 \cos (π x + y) (π + y')$

Step 1.4

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

$y' = 3 \cos (π x + y) (π + y')$

Step 1.5

Solve for $y'$ .

Tap for more steps...

Step 1.5.1

Simplify the right side.

Tap for more steps...

Step 1.5.1.1

Simplify $3 \cos (π x + y) (π + y')$ .

Tap for more steps...

Step 1.5.1.1.1

Apply the distributive property.

$y' = 3 \cos (π x + y) π + 3 \cos (π x + y) y'$

Step 1.5.1.1.2

Reorder factors in $3 \cos (π x + y) π + 3 \cos (π x + y) y'$ .

$y' = 3 π \cos (π x + y) + 3 y' \cos (π x + y)$

Step 1.5.2

Subtract $3 y' \cos (π x + y)$ from both sides of the equation.

$y' - 3 y' \cos (π x + y) = 3 π \cos (π x + y)$

Step 1.5.3

Factor $y'$ out of $y' - 3 y' \cos (π x + y)$ .

Tap for more steps...

Step 1.5.3.1

Factor $y'$ out of ${y'}^{1}$ .

$y' \cdot 1 - 3 y' \cos (π x + y) = 3 π \cos (π x + y)$

Step 1.5.3.2

Factor $y'$ out of $- 3 y' \cos (π x + y)$ .

$y' \cdot 1 + y' (- 3 \cos (π x + y)) = 3 π \cos (π x + y)$

Step 1.5.3.3

Factor $y'$ out of $y' \cdot 1 + y' (- 3 \cos (π x + y))$ .

$y' (1 - 3 \cos (π x + y)) = 3 π \cos (π x + y)$

Step 1.5.4

Divide each term in $y' (1 - 3 \cos (π x + y)) = 3 π \cos (π x + y)$ by $1 - 3 \cos (π x + y)$ and simplify.

Tap for more steps...

Step 1.5.4.1

Divide each term in $y' (1 - 3 \cos (π x + y)) = 3 π \cos (π x + y)$ by $1 - 3 \cos (π x + y)$ .

$\frac{y' (1 - 3 \cos (π x + y))}{1 - 3 \cos (π x + y)} = \frac{3 π \cos (π x + y)}{1 - 3 \cos (π x + y)}$

Step 1.5.4.2

Simplify the left side.

Tap for more steps...

Step 1.5.4.2.1

Cancel the common factor of $1 - 3 \cos (π x + y)$ .

Tap for more steps...

Step 1.5.4.2.1.1

Cancel the common factor.

$\frac{y' (1 - 3 \cos (π x + y))}{1 - 3 \cos (π x + y)} = \frac{3 π \cos (π x + y)}{1 - 3 \cos (π x + y)}$

Step 1.5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{3 π \cos (π x + y)}{1 - 3 \cos (π x + y)}$

Step 1.6

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

$\frac{d y}{d x} = \frac{3 π \cos (π x + y)}{1 - 3 \cos (π x + y)}$

Step 1.7

Evaluate at $x = 1$ and $y = 0$ .

Tap for more steps...

Step 1.7.1

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

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

Step 1.7.2

Replace the variable $y$ with $0$ in the expression.

$\frac{3 π \cos (π (1) + 0)}{1 - 3 \cos (π (1) + 0)}$

Step 1.7.3

Remove parentheses.

$\frac{3 π \cos (π (1) + 0)}{1 - 3 \cos (π (1) + 0)}$

Step 1.7.4

Simplify the numerator.

Tap for more steps...

Step 1.7.4.1

Multiply $π$ by $1$ .

$\frac{3 π \cos (π + 0)}{1 - 3 \cos (π \cdot 1 + 0)}$

Step 1.7.4.2

Add $π$ and $0$ .

$\frac{3 π \cos (π)}{1 - 3 \cos (π \cdot 1 + 0)}$

Step 1.7.4.3

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.

$\frac{3 π (- \cos (0))}{1 - 3 \cos (π \cdot 1 + 0)}$

Step 1.7.4.4

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

$\frac{3 π (- 1 \cdot 1)}{1 - 3 \cos (π \cdot 1 + 0)}$

Step 1.7.4.5

Multiply $- 1$ by $1$ .

$\frac{3 π \cdot - 1}{1 - 3 \cos (π \cdot 1 + 0)}$

Step 1.7.4.6

Multiply $- 1$ by $3$ .

$\frac{- 3 π}{1 - 3 \cos (π \cdot 1 + 0)}$

Step 1.7.5

Simplify the denominator.

Tap for more steps...

Step 1.7.5.1

Multiply $π$ by $1$ .

$\frac{- 3 π}{1 - 3 \cos (π + 0)}$

Step 1.7.5.2

Add $π$ and $0$ .

$\frac{- 3 π}{1 - 3 \cos (π)}$

Step 1.7.5.3

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.

$\frac{- 3 π}{1 - 3 (- \cos (0))}$

Step 1.7.5.4

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

$\frac{- 3 π}{1 - 3 (- 1 \cdot 1)}$

Step 1.7.5.5

Multiply $- 3 (- 1 \cdot 1)$ .

Tap for more steps...

Step 1.7.5.5.1

Multiply $- 1$ by $1$ .

$\frac{- 3 π}{1 - 3 \cdot - 1}$

Step 1.7.5.5.2

Multiply $- 3$ by $- 1$ .

$\frac{- 3 π}{1 + 3}$

Step 1.7.5.6

Add $1$ and $3$ .

$\frac{- 3 π}{4}$

Step 1.7.6

Move the negative in front of the fraction.

$- \frac{3 π}{4}$

Step 2

The normal line is perpendicular to the tangent line. Take the negative reciprocal of the slope of the tangent line to find the slope of the normal line.

$\frac{4}{3 π}$

Step 3

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

Tap for more steps...

Step 3.1

Use the slope $\frac{4}{3 π}$ and a given point $(1, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = \frac{4}{3 π} \cdot (x - (1))$

Step 3.2

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

$y + 0 = \frac{4}{3 π} \cdot (x - 1)$

Step 3.3

Solve for $y$ .

Tap for more steps...

Step 3.3.1

Add $y$ and $0$ .

$y = \frac{4}{3 π} \cdot (x - 1)$

Step 3.3.2

Simplify $\frac{4}{3 π} \cdot (x - 1)$ .

Tap for more steps...

Step 3.3.2.1

Apply the distributive property.

$y = \frac{4}{3 π} x + \frac{4}{3 π} \cdot - 1$

Step 3.3.2.2

Combine $\frac{4}{3 π}$ and $x$ .

$y = \frac{4 x}{3 π} + \frac{4}{3 π} \cdot - 1$

Step 3.3.2.3

Multiply $\frac{4}{3 π} \cdot - 1$ .

Tap for more steps...

Step 3.3.2.3.1

Combine $\frac{4}{3 π}$ and $- 1$ .

$y = \frac{4 x}{3 π} + \frac{4 \cdot - 1}{3 π}$

Step 3.3.2.3.2

Multiply $4$ by $- 1$ .

$y = \frac{4 x}{3 π} + \frac{- 4}{3 π}$

Step 3.3.2.4

Move the negative in front of the fraction.

$y = \frac{4 x}{3 π} - \frac{4}{3 π}$

Step 3.3.3

Reorder terms.

$y = \frac{4}{3 π} x - \frac{4}{3 π}$

Step 4