Calculus Examples

$y^{2} - x^{3} = 9$ , $(3, - 6)$

Step 1

Find the first derivative and evaluate at $x = 3$ and $y = - 6$ to find the slope of the tangent line.

Tap for more steps...

Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (y^{2} - x^{3}) = \frac{d}{d x} (9)$

Step 1.2

Differentiate the left side of the equation.

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

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

Tap for more steps...

Step 1.2.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

Tap for more steps...

Step 1.2.2.1.1

To apply the Chain Rule, set $u$ as $y$ .

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

Step 1.2.2.1.2

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

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

Step 1.2.2.1.3

Replace all occurrences of $u$ with $y$ .

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

Step 1.2.2.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

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

Step 1.2.3

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

Tap for more steps...

Step 1.2.3.1

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

$2 y y' - \frac{d}{d x} [x^{3}]$

Step 1.2.3.2

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

$2 y y' - (3 x^{2})$

Step 1.2.3.3

Multiply $3$ by $- 1$ .

$2 y y' - 3 x^{2}$

Step 1.2.4

Reorder terms.

$- 3 x^{2} + 2 y y'$

Step 1.3

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$0$

Step 1.4

Reform the equation by setting the left side equal to the right side.

$- 3 x^{2} + 2 y y' = 0$

Step 1.5

Solve for $y'$ .

Tap for more steps...

Step 1.5.1

Add $3 x^{2}$ to both sides of the equation.

$2 y y' = 3 x^{2}$

Step 1.5.2

Divide each term in $2 y y' = 3 x^{2}$ by $2 y$ and simplify.

Tap for more steps...

Step 1.5.2.1

Divide each term in $2 y y' = 3 x^{2}$ by $2 y$ .

$\frac{2 y y'}{2 y} = \frac{3 x^{2}}{2 y}$

Step 1.5.2.2

Simplify the left side.

Tap for more steps...

Step 1.5.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{2 y y'}{2 y} = \frac{3 x^{2}}{2 y}$

Step 1.5.2.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{3 x^{2}}{2 y}$

Step 1.5.2.2.2

Cancel the common factor of $y$ .

Tap for more steps...

Step 1.5.2.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{3 x^{2}}{2 y}$

Step 1.5.2.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{3 x^{2}}{2 y}$

Step 1.6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{3 x^{2}}{2 y}$

Step 1.7

Evaluate at $x = 3$ and $y = - 6$ .

Tap for more steps...

Step 1.7.1

Replace the variable $x$ with $3$ in the expression.

$\frac{3 {(3)}^{2}}{2 y}$

Step 1.7.2

Replace the variable $y$ with $- 6$ in the expression.

$\frac{3 {(3)}^{2}}{2 (- 6)}$

Step 1.7.3

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

Tap for more steps...

Step 1.7.3.1

Multiply $3$ by ${(3)}^{2}$ .

Tap for more steps...

Step 1.7.3.1.1

Raise $3$ to the power of $1$ .

$\frac{3^{1} {(3)}^{2}}{2 (- 6)}$

Step 1.7.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3^{1 + 2}}{2 (- 6)}$

Step 1.7.3.2

Add $1$ and $2$ .

$\frac{3^{3}}{2 (- 6)}$

Step 1.7.4

Simplify the expression.

Tap for more steps...

Step 1.7.4.1

Raise $3$ to the power of $3$ .

$\frac{27}{2 (- 6)}$

Step 1.7.4.2

Multiply $2$ by $- 6$ .

$\frac{27}{- 12}$

Step 1.7.5

Cancel the common factor of $27$ and $- 12$ .

Tap for more steps...

Step 1.7.5.1

Factor $3$ out of $27$ .

$\frac{3 (9)}{- 12}$

Step 1.7.5.2

Cancel the common factors.

Tap for more steps...

Step 1.7.5.2.1

Factor $3$ out of $- 12$ .

$\frac{3 \cdot 9}{3 \cdot - 4}$

Step 1.7.5.2.2

Cancel the common factor.

$\frac{3 \cdot 9}{3 \cdot - 4}$

Step 1.7.5.2.3

Rewrite the expression.

$\frac{9}{- 4}$

Step 1.7.6

Move the negative in front of the fraction.

$- \frac{9}{4}$

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 $- \frac{9}{4}$ and a given point $(3, - 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 6) = - \frac{9}{4} \cdot (x - (3))$

Step 2.2

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

$y + 6 = - \frac{9}{4} \cdot (x - 3)$

Step 2.3

Solve for $y$ .

Tap for more steps...

Step 2.3.1

Simplify $- \frac{9}{4} \cdot (x - 3)$ .

Tap for more steps...

Step 2.3.1.1

Rewrite.

$y + 6 = 0 + 0 - \frac{9}{4} \cdot (x - 3)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 6 = - \frac{9}{4} \cdot (x - 3)$

Step 2.3.1.3

Apply the distributive property.

$y + 6 = - \frac{9}{4} x - \frac{9}{4} \cdot - 3$

Step 2.3.1.4

Combine $x$ and $\frac{9}{4}$ .

$y + 6 = - \frac{x \cdot 9}{4} - \frac{9}{4} \cdot - 3$

Step 2.3.1.5

Multiply $- \frac{9}{4} \cdot - 3$ .

Tap for more steps...

Step 2.3.1.5.1

Multiply $- 3$ by $- 1$ .

$y + 6 = - \frac{x \cdot 9}{4} + 3 (\frac{9}{4})$

Step 2.3.1.5.2

Combine $3$ and $\frac{9}{4}$ .

$y + 6 = - \frac{x \cdot 9}{4} + \frac{3 \cdot 9}{4}$

Step 2.3.1.5.3

Multiply $3$ by $9$ .

$y + 6 = - \frac{x \cdot 9}{4} + \frac{27}{4}$

Step 2.3.1.6

Move $9$ to the left of $x$ .

$y + 6 = - \frac{9 x}{4} + \frac{27}{4}$

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

Subtract $6$ from both sides of the equation.

$y = - \frac{9 x}{4} + \frac{27}{4} - 6$

Step 2.3.2.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = - \frac{9 x}{4} + \frac{27}{4} - 6 \cdot \frac{4}{4}$

Step 2.3.2.3

Combine $- 6$ and $\frac{4}{4}$ .

$y = - \frac{9 x}{4} + \frac{27}{4} + \frac{- 6 \cdot 4}{4}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = - \frac{9 x}{4} + \frac{27 - 6 \cdot 4}{4}$

Step 2.3.2.5

Simplify the numerator.

Tap for more steps...

Step 2.3.2.5.1

Multiply $- 6$ by $4$ .

$y = - \frac{9 x}{4} + \frac{27 - 24}{4}$

Step 2.3.2.5.2

Subtract $24$ from $27$ .

$y = - \frac{9 x}{4} + \frac{3}{4}$

Step 2.3.3

Write in $y = m x + b$ form.

Tap for more steps...

Step 2.3.3.1

Reorder terms.

$y = - (\frac{9}{4} x) + \frac{3}{4}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{9}{4} x + \frac{3}{4}$

Step 3