Calculus Examples

$y = 1 - 6 x^{2}$ at, $(- 4, - 95)$

Step 1

Find the first derivative and evaluate at $x = - 4$ and $y = - 95$ 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 $1 - 6 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 6 x^{2}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- 6 x^{2}]$

Step 1.1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 6 x^{2}]$

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

$0 - 6 \frac{d}{d x} [x^{2}]$

Step 1.2.2

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

$0 - 6 (2 x)$

Step 1.2.3

Multiply $2$ by $- 6$ .

$0 - 12 x$

Step 1.3

Subtract $12 x$ from $0$ .

$- 12 x$

Step 1.4

Evaluate the derivative at $x = - 4$ .

$- 12 \cdot - 4$

Step 1.5

Multiply $- 12$ by $- 4$ .

$48$

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 $48$ and a given point $(- 4, - 95)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 95) = 48 \cdot (x - (- 4))$

Step 2.2

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

$y + 95 = 48 \cdot (x + 4)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $48 \cdot (x + 4)$ .

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Step 2.3.1.1

Rewrite.

$y + 95 = 0 + 0 + 48 \cdot (x + 4)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 95 = 48 \cdot (x + 4)$

Step 2.3.1.3

Apply the distributive property.

$y + 95 = 48 x + 48 \cdot 4$

Step 2.3.1.4

Multiply $48$ by $4$ .

$y + 95 = 48 x + 192$

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

Subtract $95$ from both sides of the equation.

$y = 48 x + 192 - 95$

Step 2.3.2.2

Subtract $95$ from $192$ .

$y = 48 x + 97$

Step 3