Calculus Examples

$f (x) = x^{4} - 20 x^{2} + 64$ ; $x = 2$

Step 1

Find the corresponding $y$ -value to $x = 2$ .

Tap for more steps...

Step 1.1

Substitute $2$ in for $x$ .

$y = {(2)}^{4} - 20 {(2)}^{2} + 64$

Step 1.2

Solve for $y$ .

Tap for more steps...

Step 1.2.1

Remove parentheses.

$y = 2^{4} - 20 {(2)}^{2} + 64$

Step 1.2.2

Remove parentheses.

$y = 2^{4} - 20 \cdot 2^{2} + 64$

Step 1.2.3

Remove parentheses.

$y = {(2)}^{4} - 20 {(2)}^{2} + 64$

Step 1.2.4

Simplify ${(2)}^{4} - 20 {(2)}^{2} + 64$ .

Tap for more steps...

Step 1.2.4.1

Simplify each term.

Tap for more steps...

Step 1.2.4.1.1

Raise $2$ to the power of $4$ .

$y = 16 - 20 {(2)}^{2} + 64$

Step 1.2.4.1.2

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

$y = 16 - 20 \cdot 4 + 64$

Step 1.2.4.1.3

Multiply $- 20$ by $4$ .

$y = 16 - 80 + 64$

Step 1.2.4.2

Simplify by adding and subtracting.

Tap for more steps...

Step 1.2.4.2.1

Subtract $80$ from $16$ .

$y = - 64 + 64$

Step 1.2.4.2.2

Add $- 64$ and $64$ .

$y = 0$

Step 2

Find the first derivative and evaluate at $x = 2$ and $y = 0$ to find the slope of the tangent line.

Tap for more steps...

Step 2.1

Differentiate.

Tap for more steps...

Step 2.1.1

By the Sum Rule, the derivative of $x^{4} - 20 x^{2} + 64$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 20 x^{2}] + \frac{d}{d x} [64]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 20 x^{2}] + \frac{d}{d x} [64]$

Step 2.1.2

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

$4 x^{3} + \frac{d}{d x} [- 20 x^{2}] + \frac{d}{d x} [64]$

Step 2.2

Evaluate $\frac{d}{d x} [- 20 x^{2}]$ .

Tap for more steps...

Step 2.2.1

Since $- 20$ is constant with respect to $x$ , the derivative of $- 20 x^{2}$ with respect to $x$ is $- 20 \frac{d}{d x} [x^{2}]$ .

$4 x^{3} - 20 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [64]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$4 x^{3} - 20 (2 x) + \frac{d}{d x} [64]$

Step 2.2.3

Multiply $2$ by $- 20$ .

$4 x^{3} - 40 x + \frac{d}{d x} [64]$

Step 2.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.3.1

Since $64$ is constant with respect to $x$ , the derivative of $64$ with respect to $x$ is $0$ .

$4 x^{3} - 40 x + 0$

Step 2.3.2

Add $4 x^{3} - 40 x$ and $0$ .

$4 x^{3} - 40 x$

Step 2.4

Evaluate the derivative at $x = 2$ .

$4 {(2)}^{3} - 40 \cdot 2$

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

Simplify each term.

Tap for more steps...

Step 2.5.1.1

Raise $2$ to the power of $3$ .

$4 \cdot 8 - 40 \cdot 2$

Step 2.5.1.2

Multiply $4$ by $8$ .

$32 - 40 \cdot 2$

Step 2.5.1.3

Multiply $- 40$ by $2$ .

$32 - 80$

Step 2.5.2

Subtract $80$ from $32$ .

$- 48$

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 $- 48$ and a given point $(2, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = - 48 \cdot (x - (2))$

Step 3.2

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

$y + 0 = - 48 \cdot (x - 2)$

Step 3.3

Solve for $y$ .

Tap for more steps...

Step 3.3.1

Add $y$ and $0$ .

$y = - 48 \cdot (x - 2)$

Step 3.3.2

Simplify $- 48 \cdot (x - 2)$ .

Tap for more steps...

Step 3.3.2.1

Apply the distributive property.

$y = - 48 x - 48 \cdot - 2$

Step 3.3.2.2

Multiply $- 48$ by $- 2$ .

$y = - 48 x + 96$

Step 4