Calculus Examples

$y = x^{4} + 8 x^{2} - x$ , $(1, 8)$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = 8$ 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 $x^{4} + 8 x^{2} - x$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [8 x^{2}] + \frac{d}{d x} [- x]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [8 x^{2}] + \frac{d}{d x} [- x]$

Step 1.1.2

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

$4 x^{3} + \frac{d}{d x} [8 x^{2}] + \frac{d}{d x} [- x]$

Step 1.2

Evaluate $\frac{d}{d x} [8 x^{2}]$ .

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Step 1.2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{2}$ with respect to $x$ is $8 \frac{d}{d x} [x^{2}]$ .

$4 x^{3} + 8 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x]$

Step 1.2.2

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

$4 x^{3} + 8 (2 x) + \frac{d}{d x} [- x]$

Step 1.2.3

Multiply $2$ by $8$ .

$4 x^{3} + 16 x + \frac{d}{d x} [- x]$

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

$4 x^{3} + 16 x - \frac{d}{d x} [x]$

Step 1.3.2

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

$4 x^{3} + 16 x - 1 \cdot 1$

Step 1.3.3

Multiply $- 1$ by $1$ .

$4 x^{3} + 16 x - 1$

Step 1.4

Evaluate the derivative at $x = 1$ .

$4 {(1)}^{3} + 16 (1) - 1$

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.

$4 \cdot 1 + 16 (1) - 1$

Step 1.5.1.2

Multiply $4$ by $1$ .

$4 + 16 (1) - 1$

Step 1.5.1.3

Multiply $16$ by $1$ .

$4 + 16 - 1$

Step 1.5.2

Simplify by adding and subtracting.

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Step 1.5.2.1

Add $4$ and $16$ .

$20 - 1$

Step 1.5.2.2

Subtract $1$ from $20$ .

$19$

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 $19$ and a given point $(1, 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (8) = 19 \cdot (x - (1))$

Step 2.2

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

$y - 8 = 19 \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $19 \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 8 = 0 + 0 + 19 \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 8 = 19 \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y - 8 = 19 x + 19 \cdot - 1$

Step 2.3.1.4

Multiply $19$ by $- 1$ .

$y - 8 = 19 x - 19$

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 $8$ to both sides of the equation.

$y = 19 x - 19 + 8$

Step 2.3.2.2

Add $- 19$ and $8$ .

$y = 19 x - 11$

Step 3