Calculus Examples

$y = x^{4} + 5 x^{2} - x$ ,

$(1, 5)$

Step 1

Find the first derivative and evaluate at

$x = 1$ and

$y = 5$ to find the slope of the tangent line.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of

$x^{4} + 5 x^{2} - x$ with respect to

$x$ is

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- x]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- x]$

Step 1.1.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 4$ .

$4 x^{3} + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- x]$

Step 1.2

Evaluate

$\frac{d}{d x} [5 x^{2}]$ .

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Step 1.2.1

Since

$5$ is constant with respect to

$x$ , the derivative of

$5 x^{2}$ with respect to

$x$ is

$5 \frac{d}{d x} [x^{2}]$ .

$4 x^{3} + 5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x]$

Step 1.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 2$ .

$4 x^{3} + 5 (2 x) + \frac{d}{d x} [- x]$

Step 1.2.3

Multiply

$2$ by

$5$ .

$4 x^{3} + 10 x + \frac{d}{d x} [- x]$

Step 1.3

Evaluate

$\frac{d}{d x} [- x]$ .

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Step 1.3.1

Since

$- 1$ is constant with respect to

$x$ , the derivative of

$- x$ with respect to

$x$ is

$- \frac{d}{d x} [x]$ .

$4 x^{3} + 10 x - \frac{d}{d x} [x]$

Step 1.3.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$4 x^{3} + 10 x - 1 \cdot 1$

Step 1.3.3

Multiply

$- 1$ by

$1$ .

$4 x^{3} + 10 x - 1$

Step 1.4

Evaluate the derivative at

$x = 1$ .

$4 {(1)}^{3} + 10 (1) - 1$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

One to any power is one.

$4 \cdot 1 + 10 (1) - 1$

Step 1.5.1.2

Multiply

$4$ by

$1$ .

$4 + 10 (1) - 1$

Step 1.5.1.3

Multiply

$10$ by

$1$ .

$4 + 10 - 1$

Step 1.5.2

Simplify by adding and subtracting.

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Step 1.5.2.1

Add

$4$ and

$10$ .

$14 - 1$

Step 1.5.2.2

Subtract

$1$ from

$14$ .

$13$

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

$13$ and a given point

$(1, 5)$ 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

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = 13 \cdot (x - (1))$

Step 2.2

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

$y - 5 = 13 \cdot (x - 1)$

Step 2.3

Solve for

$y$ .

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Step 2.3.1

Simplify

$13 \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 5 = 0 + 0 + 13 \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 5 = 13 \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y - 5 = 13 x + 13 \cdot - 1$

Step 2.3.1.4

Multiply

$13$ by

$- 1$ .

$y - 5 = 13 x - 13$

Step 2.3.2

Move all terms not containing

$y$ to the right side of the equation.

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Step 2.3.2.1

Add

$5$ to both sides of the equation.

$y = 13 x - 13 + 5$

Step 2.3.2.2

Add

$- 13$ and

$5$ .

$y = 13 x - 8$

Step 3