Calculus Examples

$y = 6 x^{2} - x^{3}$ , $(1, 5)$

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

By the Sum Rule, the derivative of $6 x^{2} - x^{3}$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- x^{3}]$ .

$\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- x^{3}]$

Step 1.2

Evaluate $\frac{d}{d x} [6 x^{2}]$ .

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Step 1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x^{3}]$

Step 1.2.2

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

$6 (2 x) + \frac{d}{d x} [- x^{3}]$

Step 1.2.3

Multiply $2$ by $6$ .

$12 x + \frac{d}{d x} [- x^{3}]$

Step 1.3

Evaluate $\frac{d}{d x} [- x^{3}]$ .

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Step 1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

$12 x - \frac{d}{d x} [x^{3}]$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$12 x - (3 x^{2})$

Step 1.3.3

Multiply $3$ by $- 1$ .

$12 x - 3 x^{2}$

Step 1.4

Reorder terms.

$- 3 x^{2} + 12 x$

Step 1.5

Evaluate the derivative at $x = 1$ .

$- 3 {(1)}^{2} + 12 (1)$

Step 1.6

Simplify.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

One to any power is one.

$- 3 \cdot 1 + 12 (1)$

Step 1.6.1.2

Multiply $- 3$ by $1$ .

$- 3 + 12 (1)$

Step 1.6.1.3

Multiply $12$ by $1$ .

$- 3 + 12$

Step 1.6.2

Add $- 3$ and $12$ .

$9$

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 $9$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = 9 \cdot (x - (1))$

Step 2.2

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

$y - 5 = 9 \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $9 \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 5 = 0 + 0 + 9 \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 5 = 9 \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y - 5 = 9 x + 9 \cdot - 1$

Step 2.3.1.4

Multiply $9$ by $- 1$ .

$y - 5 = 9 x - 9$

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.

$y = 9 x - 9 + 5$

Step 2.3.2.2

Add $- 9$ and $5$ .

$y = 9 x - 4$

Step 3