Calculus Examples

$16 x^{3} + y^{3} = 3$ , $(2, - 5)$

Step 1

Find the first derivative and evaluate at $x = 2$ and $y = - 5$ to find the slope of the tangent line.

Tap for more steps...

Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (16 x^{3} + y^{3}) = \frac{d}{d x} (3)$

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 $16 x^{3} + y^{3}$ with respect to $x$ is $\frac{d}{d x} [16 x^{3}] + \frac{d}{d x} [y^{3}]$ .

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

Step 1.2.2

Evaluate $\frac{d}{d x} [16 x^{3}]$ .

Tap for more steps...

Step 1.2.2.1

Since $16$ is constant with respect to $x$ , the derivative of $16 x^{3}$ with respect to $x$ is $16 \frac{d}{d x} [x^{3}]$ .

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

Step 1.2.2.2

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

$16 (3 x^{2}) + \frac{d}{d x} [y^{3}]$

Step 1.2.2.3

Multiply $3$ by $16$ .

$48 x^{2} + \frac{d}{d x} [y^{3}]$

Step 1.2.3

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

Tap for more steps...

Step 1.2.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = y$ .

Tap for more steps...

Step 1.2.3.1.1

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

$48 x^{2} + \frac{d}{d u} [u^{3}] \frac{d}{d x} [y]$

Step 1.2.3.1.2

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

$48 x^{2} + 3 u^{2} \frac{d}{d x} [y]$

Step 1.2.3.1.3

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

$48 x^{2} + 3 y^{2} \frac{d}{d x} [y]$

Step 1.2.3.2

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

$48 x^{2} + 3 y^{2} y'$

Step 1.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0$

Step 1.4

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

$48 x^{2} + 3 y^{2} y' = 0$

Step 1.5

Solve for $y'$ .

Tap for more steps...

Step 1.5.1

Subtract $48 x^{2}$ from both sides of the equation.

$3 y^{2} y' = - 48 x^{2}$

Step 1.5.2

Divide each term in $3 y^{2} y' = - 48 x^{2}$ by $3 y^{2}$ and simplify.

Tap for more steps...

Step 1.5.2.1

Divide each term in $3 y^{2} y' = - 48 x^{2}$ by $3 y^{2}$ .

$\frac{3 y^{2} y'}{3 y^{2}} = \frac{- 48 x^{2}}{3 y^{2}}$

Step 1.5.2.2

Simplify the left side.

Tap for more steps...

Step 1.5.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{3 y^{2} y'}{3 y^{2}} = \frac{- 48 x^{2}}{3 y^{2}}$

Step 1.5.2.2.1.2

Rewrite the expression.

$\frac{y^{2} y'}{y^{2}} = \frac{- 48 x^{2}}{3 y^{2}}$

Step 1.5.2.2.2

Cancel the common factor of $y^{2}$ .

Tap for more steps...

Step 1.5.2.2.2.1

Cancel the common factor.

$\frac{y^{2} y'}{y^{2}} = \frac{- 48 x^{2}}{3 y^{2}}$

Step 1.5.2.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 48 x^{2}}{3 y^{2}}$

Step 1.5.2.3

Simplify the right side.

Tap for more steps...

Step 1.5.2.3.1

Cancel the common factor of $- 48$ and $3$ .

Tap for more steps...

Step 1.5.2.3.1.1

Factor $3$ out of $- 48 x^{2}$ .

$y' = \frac{3 (- 16 x^{2})}{3 y^{2}}$

Step 1.5.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 1.5.2.3.1.2.1

Factor $3$ out of $3 y^{2}$ .

$y' = \frac{3 (- 16 x^{2})}{3 (y^{2})}$

Step 1.5.2.3.1.2.2

Cancel the common factor.

$y' = \frac{3 (- 16 x^{2})}{3 y^{2}}$

Step 1.5.2.3.1.2.3

Rewrite the expression.

$y' = \frac{- 16 x^{2}}{y^{2}}$

Step 1.5.2.3.2

Move the negative in front of the fraction.

$y' = - \frac{16 x^{2}}{y^{2}}$

Step 1.6

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

$\frac{d y}{d x} = - \frac{16 x^{2}}{y^{2}}$

Step 1.7

Evaluate at $x = 2$ and $y = - 5$ .

Tap for more steps...

Step 1.7.1

Replace the variable $x$ with $2$ in the expression.

$- \frac{16 {(2)}^{2}}{y^{2}}$

Step 1.7.2

Replace the variable $y$ with $- 5$ in the expression.

$- \frac{16 {(2)}^{2}}{{(- 5)}^{2}}$

Step 1.7.3

Simplify the numerator.

Tap for more steps...

Step 1.7.3.1

Rewrite $16$ as $2^{4}$ .

$- \frac{2^{4} \cdot 2^{2}}{{(- 5)}^{2}}$

Step 1.7.3.2

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

$- \frac{2^{4 + 2}}{{(- 5)}^{2}}$

Step 1.7.3.3

Add $4$ and $2$ .

$- \frac{2^{6}}{{(- 5)}^{2}}$

Step 1.7.4

Evaluate the exponents.

Tap for more steps...

Step 1.7.4.1

Raise $- 5$ to the power of $2$ .

$- \frac{2^{6}}{25}$

Step 1.7.4.2

Raise $2$ to the power of $6$ .

$- \frac{64}{25}$

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 $- \frac{64}{25}$ and a given point $(2, - 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 5) = - \frac{64}{25} \cdot (x - (2))$

Step 2.2

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

$y + 5 = - \frac{64}{25} \cdot (x - 2)$

Step 2.3

Solve for $y$ .

Tap for more steps...

Step 2.3.1

Simplify $- \frac{64}{25} \cdot (x - 2)$ .

Tap for more steps...

Step 2.3.1.1

Rewrite.

$y + 5 = 0 + 0 - \frac{64}{25} \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 5 = - \frac{64}{25} \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

$y + 5 = - \frac{64}{25} x - \frac{64}{25} \cdot - 2$

Step 2.3.1.4

Combine $x$ and $\frac{64}{25}$ .

$y + 5 = - \frac{x \cdot 64}{25} - \frac{64}{25} \cdot - 2$

Step 2.3.1.5

Multiply $- \frac{64}{25} \cdot - 2$ .

Tap for more steps...

Step 2.3.1.5.1

Multiply $- 2$ by $- 1$ .

$y + 5 = - \frac{x \cdot 64}{25} + 2 (\frac{64}{25})$

Step 2.3.1.5.2

Combine $2$ and $\frac{64}{25}$ .

$y + 5 = - \frac{x \cdot 64}{25} + \frac{2 \cdot 64}{25}$

Step 2.3.1.5.3

Multiply $2$ by $64$ .

$y + 5 = - \frac{x \cdot 64}{25} + \frac{128}{25}$

Step 2.3.1.6

Move $64$ to the left of $x$ .

$y + 5 = - \frac{64 x}{25} + \frac{128}{25}$

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 $5$ from both sides of the equation.

$y = - \frac{64 x}{25} + \frac{128}{25} - 5$

Step 2.3.2.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$y = - \frac{64 x}{25} + \frac{128}{25} - 5 \cdot \frac{25}{25}$

Step 2.3.2.3

Combine $- 5$ and $\frac{25}{25}$ .

$y = - \frac{64 x}{25} + \frac{128}{25} + \frac{- 5 \cdot 25}{25}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = - \frac{64 x}{25} + \frac{128 - 5 \cdot 25}{25}$

Step 2.3.2.5

Simplify the numerator.

Tap for more steps...

Step 2.3.2.5.1

Multiply $- 5$ by $25$ .

$y = - \frac{64 x}{25} + \frac{128 - 125}{25}$

Step 2.3.2.5.2

Subtract $125$ from $128$ .

$y = - \frac{64 x}{25} + \frac{3}{25}$

Step 2.3.3

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

Tap for more steps...

Step 2.3.3.1

Reorder terms.

$y = - (\frac{64}{25} x) + \frac{3}{25}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{64}{25} x + \frac{3}{25}$

Step 3