Calculus Examples

$f (x) = x^{4} - 25 x^{2} + 144$ ; $x = 2$

Step 1

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

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Step 1.1

Substitute $2$ in for $x$ .

$y = {(2)}^{4} - 25 {(2)}^{2} + 144$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = 2^{4} - 25 {(2)}^{2} + 144$

Step 1.2.2

Remove parentheses.

$y = 2^{4} - 25 \cdot 2^{2} + 144$

Step 1.2.3

Remove parentheses.

$y = {(2)}^{4} - 25 {(2)}^{2} + 144$

Step 1.2.4

Simplify ${(2)}^{4} - 25 {(2)}^{2} + 144$ .

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Step 1.2.4.1

Simplify each term.

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Step 1.2.4.1.1

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

$y = 16 - 25 {(2)}^{2} + 144$

Step 1.2.4.1.2

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

$y = 16 - 25 \cdot 4 + 144$

Step 1.2.4.1.3

Multiply $- 25$ by $4$ .

$y = 16 - 100 + 144$

Step 1.2.4.2

Simplify by adding and subtracting.

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Step 1.2.4.2.1

Subtract $100$ from $16$ .

$y = - 84 + 144$

Step 1.2.4.2.2

Add $- 84$ and $144$ .

$y = 60$

Step 2

Find the first derivative and evaluate at $x = 2$ and $y = 60$ to find the slope of the tangent line.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{4} - 25 x^{2} + 144$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 25 x^{2}] + \frac{d}{d x} [144]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 25 x^{2}] + \frac{d}{d x} [144]$

Step 2.1.2

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

$4 x^{3} + \frac{d}{d x} [- 25 x^{2}] + \frac{d}{d x} [144]$

Step 2.2

Evaluate $\frac{d}{d x} [- 25 x^{2}]$ .

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Step 2.2.1

Since $- 25$ is constant with respect to $x$ , the derivative of $- 25 x^{2}$ with respect to $x$ is $- 25 \frac{d}{d x} [x^{2}]$ .

$4 x^{3} - 25 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [144]$

Step 2.2.2

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

$4 x^{3} - 25 (2 x) + \frac{d}{d x} [144]$

Step 2.2.3

Multiply $2$ by $- 25$ .

$4 x^{3} - 50 x + \frac{d}{d x} [144]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $144$ is constant with respect to $x$ , the derivative of $144$ with respect to $x$ is $0$ .

$4 x^{3} - 50 x + 0$

Step 2.3.2

Add $4 x^{3} - 50 x$ and $0$ .

$4 x^{3} - 50 x$

Step 2.4

Evaluate the derivative at $x = 2$ .

$4 {(2)}^{3} - 50 \cdot 2$

Step 2.5

Simplify.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

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

$4 \cdot 8 - 50 \cdot 2$

Step 2.5.1.2

Multiply $4$ by $8$ .

$32 - 50 \cdot 2$

Step 2.5.1.3

Multiply $- 50$ by $2$ .

$32 - 100$

Step 2.5.2

Subtract $100$ from $32$ .

$- 68$

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 $- 68$ and a given point $(2, 60)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (60) = - 68 \cdot (x - (2))$

Step 3.2

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

$y - 60 = - 68 \cdot (x - 2)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- 68 \cdot (x - 2)$ .

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Step 3.3.1.1

Rewrite.

$y - 60 = 0 + 0 - 68 \cdot (x - 2)$

Step 3.3.1.2

Simplify by adding zeros.

$y - 60 = - 68 \cdot (x - 2)$

Step 3.3.1.3

Apply the distributive property.

$y - 60 = - 68 x - 68 \cdot - 2$

Step 3.3.1.4

Multiply $- 68$ by $- 2$ .

$y - 60 = - 68 x + 136$

Step 3.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.2.1

Add $60$ to both sides of the equation.

$y = - 68 x + 136 + 60$

Step 3.3.2.2

Add $136$ and $60$ .

$y = - 68 x + 196$

Step 4