Calculus Examples

$f (x) = x \sin (10 x)$ at $x = π$

Step 1

Find the corresponding $y$ -value to $x = π$ .

Tap for more steps...

Step 1.1

Substitute $π$ in for $x$ .

$y = (π) \sin (10 (π))$

Step 1.2

Solve for $y$ .

Tap for more steps...

Step 1.2.1

Multiply $10$ by $π$ .

$y = π \sin (10 π)$

Step 1.2.2

Remove parentheses.

$y = (π) \sin (10 (π))$

Step 1.2.3

Simplify $(π) \sin (10 (π))$ .

Tap for more steps...

Step 1.2.3.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$y = π \sin (0)$

Step 1.2.3.2

The exact value of $\sin (0)$ is $0$ .

$y = π \cdot 0$

Step 1.2.3.3

Multiply $π$ by $0$ .

$y = 0$

Step 2

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

Tap for more steps...

Step 2.1

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 $f (x) = x$ and $g (x) = \sin (10 x)$ .

$x \frac{d}{d x} [\sin (10 x)] + \sin (10 x) \frac{d}{d x} [x]$

Step 2.2

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 $g (x) = 10 x$ .

Tap for more steps...

Step 2.2.1

To apply the Chain Rule, set $u$ as $10 x$ .

$x (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [10 x]) + \sin (10 x) \frac{d}{d x} [x]$

Step 2.2.2

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

$x (\cos (u) \frac{d}{d x} [10 x]) + \sin (10 x) \frac{d}{d x} [x]$

Step 2.2.3

Replace all occurrences of $u$ with $10 x$ .

$x (\cos (10 x) \frac{d}{d x} [10 x]) + \sin (10 x) \frac{d}{d x} [x]$

Step 2.3

Differentiate.

Tap for more steps...

Step 2.3.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$x (\cos (10 x) (10 \frac{d}{d x} [x])) + \sin (10 x) \frac{d}{d x} [x]$

Step 2.3.2

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

$x (\cos (10 x) (10 \cdot 1)) + \sin (10 x) \frac{d}{d x} [x]$

Step 2.3.3

Simplify the expression.

Tap for more steps...

Step 2.3.3.1

Multiply $10$ by $1$ .

$x (\cos (10 x) \cdot 10) + \sin (10 x) \frac{d}{d x} [x]$

Step 2.3.3.2

Move $10$ to the left of $\cos (10 x)$ .

$x (10 \cdot \cos (10 x)) + \sin (10 x) \frac{d}{d x} [x]$

Step 2.3.4

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

$x (10 \cos (10 x)) + \sin (10 x) \cdot 1$

Step 2.3.5

Simplify the expression.

Tap for more steps...

Step 2.3.5.1

Multiply $\sin (10 x)$ by $1$ .

$x (10 \cos (10 x)) + \sin (10 x)$

Step 2.3.5.2

Reorder terms.

$10 x \cos (10 x) + \sin (10 x)$

Step 2.4

Evaluate the derivative at $x = π$ .

$10 (π) \cos (10 (π)) + \sin (10 (π))$

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

Simplify each term.

Tap for more steps...

Step 2.5.1.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$10 π \cos (0) + \sin (10 (π))$

Step 2.5.1.2

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

$10 π \cdot 1 + \sin (10 (π))$

Step 2.5.1.3

Multiply $10$ by $1$ .

$10 π + \sin (10 (π))$

Step 2.5.1.4

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$10 π + \sin (0)$

Step 2.5.1.5

The exact value of $\sin (0)$ is $0$ .

$10 π + 0$

Step 2.5.2

Add $10 π$ and $0$ .

$10 π$

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 $10 π$ and a given point $(π, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = 10 π \cdot (x - (π))$

Step 3.2

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

$y + 0 = 10 π \cdot (x - π)$

Step 3.3

Solve for $y$ .

Tap for more steps...

Step 3.3.1

Add $y$ and $0$ .

$y = 10 π \cdot (x - π)$

Step 3.3.2

Simplify $10 π \cdot (x - π)$ .

Tap for more steps...

Step 3.3.2.1

Apply the distributive property.

$y = 10 π x + 10 π (- π)$

Step 3.3.2.2

Multiply $10 π (- π)$ .

Tap for more steps...

Step 3.3.2.2.1

Multiply $- 1$ by $10$ .

$y = 10 π x - 10 π π$

Step 3.3.2.2.2

Raise $π$ to the power of $1$ .

$y = 10 π x - 10 (π^{1} π)$

Step 3.3.2.2.3

Raise $π$ to the power of $1$ .

$y = 10 π x - 10 (π^{1} π^{1})$

Step 3.3.2.2.4

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

$y = 10 π x - 10 π^{1 + 1}$

Step 3.3.2.2.5

Add $1$ and $1$ .

$y = 10 π x - 10 π^{2}$

Step 4