Calculus Examples

$y = - \frac{1}{x^{3}}$ , $(- 1, 1)$

Step 1

Find the first derivative and evaluate at $x = - 1$ and $y = 1$ to find the slope of the tangent line.

Tap for more steps...

Step 1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x^{3}}$ .

$- \frac{d}{d x} [\frac{1}{x^{3}}] + \frac{1}{x^{3}} \frac{d}{d x} [- 1]$

Step 1.2

Differentiate.

Tap for more steps...

Step 1.2.1

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$- \frac{d}{d x} [{(x^{3})}^{- 1}] + \frac{1}{x^{3}} \frac{d}{d x} [- 1]$

Step 1.2.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

Tap for more steps...

Step 1.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{d}{d x} [x^{3 \cdot - 1}] + \frac{1}{x^{3}} \frac{d}{d x} [- 1]$

Step 1.2.2.2

Multiply $3$ by $- 1$ .

$- \frac{d}{d x} [x^{- 3}] + \frac{1}{x^{3}} \frac{d}{d x} [- 1]$

Step 1.2.3

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

$- (- 3 x^{- 4}) + \frac{1}{x^{3}} \frac{d}{d x} [- 1]$

Step 1.2.4

Multiply $- 3$ by $- 1$ .

$3 x^{- 4} + \frac{1}{x^{3}} \frac{d}{d x} [- 1]$

Step 1.2.5

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

$3 x^{- 4} + \frac{1}{x^{3}} \cdot 0$

Step 1.2.6

Simplify the expression.

Tap for more steps...

Step 1.2.6.1

Multiply $\frac{1}{x^{3}}$ by $0$ .

$3 x^{- 4} + 0$

Step 1.2.6.2

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

$3 x^{- 4}$

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 \frac{1}{x^{4}}$

Step 1.3.2

Combine $3$ and $\frac{1}{x^{4}}$ .

$\frac{3}{x^{4}}$

Step 1.4

Evaluate the derivative at $x = - 1$ .

$\frac{3}{{(- 1)}^{4}}$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Raise $- 1$ to the power of $4$ .

$\frac{3}{1}$

Step 1.5.2

Divide $3$ by $1$ .

$3$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

Tap for more steps...

Step 2.1

Use the slope $3$ and a given point $(- 1, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = 3 \cdot (x - (- 1))$

Step 2.2

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

$y - 1 = 3 \cdot (x + 1)$

Step 2.3

Solve for $y$ .

Tap for more steps...

Step 2.3.1

Simplify $3 \cdot (x + 1)$ .

Tap for more steps...

Step 2.3.1.1

Rewrite.

$y - 1 = 0 + 0 + 3 \cdot (x + 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 1 = 3 \cdot (x + 1)$

Step 2.3.1.3

Apply the distributive property.

$y - 1 = 3 x + 3 \cdot 1$

Step 2.3.1.4

Multiply $3$ by $1$ .

$y - 1 = 3 x + 3$

Step 2.3.2

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

Tap for more steps...

Step 2.3.2.1

Add $1$ to both sides of the equation.

$y = 3 x + 3 + 1$

Step 2.3.2.2

Add $3$ and $1$ .

$y = 3 x + 4$

Step 3