Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (2 x^{2} + y^{3}) = \frac{d}{d x} (5 y + 3 x)$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

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

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

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 $n = 2$ .

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

Step 1.2.2.3

Multiply $2$ by $2$ .

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

Step 1.2.3

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

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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 $g (x) = y$ .

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Step 1.2.3.1.1

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

$4 x + \frac{d}{d u} [u^{3}] \frac{d}{d x} [y]$

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 $n = 3$ .

$4 x + 3 u^{2} \frac{d}{d x} [y]$

Step 1.2.3.1.3

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

$4 x + 3 y^{2} \frac{d}{d x} [y]$

Step 1.2.3.2

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

$4 x + 3 y^{2} y'$

Step 1.2.4

Reorder terms.

$3 y^{2} y' + 4 x$

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

By the Sum Rule, the derivative of $5 y + 3 x$ with respect to $x$ is $\frac{d}{d x} [5 y] + \frac{d}{d x} [3 x]$ .

$\frac{d}{d x} [5 y] + \frac{d}{d x} [3 x]$

Step 1.3.2

Evaluate $\frac{d}{d x} [5 y]$ .

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Step 1.3.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 y$ with respect to $x$ is $5 \frac{d}{d x} [y]$ .

$5 \frac{d}{d x} [y] + \frac{d}{d x} [3 x]$

Step 1.3.2.2

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

$5 y' + \frac{d}{d x} [3 x]$

Step 1.3.3

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

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Step 1.3.3.1

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

$5 y' + 3 \frac{d}{d x} [x]$

Step 1.3.3.2

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

$5 y' + 3 \cdot 1$

Step 1.3.3.3

Multiply $3$ by $1$ .

$5 y' + 3$

Step 1.4

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

$3 y^{2} y' + 4 x = 5 y' + 3$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Subtract $5 y'$ from both sides of the equation.

$3 y^{2} y' + 4 x - 5 y' = 3$

Step 1.5.2

Subtract $4 x$ from both sides of the equation.

$3 y^{2} y' - 5 y' = 3 - 4 x$

Step 1.5.3

Factor $y'$ out of $3 y^{2} y' - 5 y'$ .

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Step 1.5.3.1

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

$y' (3 y^{2}) - 5 y' = 3 - 4 x$

Step 1.5.3.2

Factor $y'$ out of $- 5 y'$ .

$y' (3 y^{2}) + y' \cdot - 5 = 3 - 4 x$

Step 1.5.3.3

Factor $y'$ out of $y' (3 y^{2}) + y' \cdot - 5$ .

$y' (3 y^{2} - 5) = 3 - 4 x$

Step 1.5.4

Divide each term in $y' (3 y^{2} - 5) = 3 - 4 x$ by $3 y^{2} - 5$ and simplify.

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Step 1.5.4.1

Divide each term in $y' (3 y^{2} - 5) = 3 - 4 x$ by $3 y^{2} - 5$ .

$\frac{y' (3 y^{2} - 5)}{3 y^{2} - 5} = \frac{3}{3 y^{2} - 5} + \frac{- 4 x}{3 y^{2} - 5}$

Step 1.5.4.2

Simplify the left side.

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Step 1.5.4.2.1

Cancel the common factor of $3 y^{2} - 5$ .

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Step 1.5.4.2.1.1

Cancel the common factor.

$\frac{y' (3 y^{2} - 5)}{3 y^{2} - 5} = \frac{3}{3 y^{2} - 5} + \frac{- 4 x}{3 y^{2} - 5}$

Step 1.5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{3}{3 y^{2} - 5} + \frac{- 4 x}{3 y^{2} - 5}$

Step 1.5.4.3

Simplify the right side.

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Step 1.5.4.3.1

Move the negative in front of the fraction.

$y' = \frac{3}{3 y^{2} - 5} - \frac{4 x}{3 y^{2} - 5}$

Step 1.5.4.3.2

Combine the numerators over the common denominator.

$y' = \frac{3 - 4 x}{3 y^{2} - 5}$

Step 1.6

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

$\frac{d y}{d x} = \frac{3 - 4 x}{3 y^{2} - 5}$

Step 1.7

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

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Step 1.7.1

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

$\frac{3 - 4 \cdot 2}{3 y^{2} - 5}$

Step 1.7.2

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

$\frac{3 - 4 \cdot 2}{3 {(2)}^{2} - 5}$

Step 1.7.3

Simplify the numerator.

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Step 1.7.3.1

Multiply $- 4$ by $2$ .

$\frac{3 - 8}{3 \cdot 2^{2} - 5}$

Step 1.7.3.2

Subtract $8$ from $3$ .

$\frac{- 5}{3 \cdot 2^{2} - 5}$

Step 1.7.4

Simplify the denominator.

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Step 1.7.4.1

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

$\frac{- 5}{3 \cdot 4 - 5}$

Step 1.7.4.2

Multiply $3$ by $4$ .

$\frac{- 5}{12 - 5}$

Step 1.7.4.3

Subtract $5$ from $12$ .

$\frac{- 5}{7}$

Step 1.7.5

Move the negative in front of the fraction.

$- \frac{5}{7}$

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 $- \frac{5}{7}$ and a given point $(2, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = - \frac{5}{7} \cdot (x - (2))$

Step 2.2

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

$y - 2 = - \frac{5}{7} \cdot (x - 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{5}{7} \cdot (x - 2)$ .

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Step 2.3.1.1

Rewrite.

$y - 2 = 0 + 0 - \frac{5}{7} \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 2 = - \frac{5}{7} \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

$y - 2 = - \frac{5}{7} x - \frac{5}{7} \cdot - 2$

Step 2.3.1.4

Combine $x$ and $\frac{5}{7}$ .

$y - 2 = - \frac{x \cdot 5}{7} - \frac{5}{7} \cdot - 2$

Step 2.3.1.5

Multiply $- \frac{5}{7} \cdot - 2$ .

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Step 2.3.1.5.1

Multiply $- 2$ by $- 1$ .

$y - 2 = - \frac{x \cdot 5}{7} + 2 (\frac{5}{7})$

Step 2.3.1.5.2

Combine $2$ and $\frac{5}{7}$ .

$y - 2 = - \frac{x \cdot 5}{7} + \frac{2 \cdot 5}{7}$

Step 2.3.1.5.3

Multiply $2$ by $5$ .

$y - 2 = - \frac{x \cdot 5}{7} + \frac{10}{7}$

Step 2.3.1.6

Move $5$ to the left of $x$ .

$y - 2 = - \frac{5 x}{7} + \frac{10}{7}$

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 $2$ to both sides of the equation.

$y = - \frac{5 x}{7} + \frac{10}{7} + 2$

Step 2.3.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = - \frac{5 x}{7} + \frac{10}{7} + 2 \cdot \frac{7}{7}$

Step 2.3.2.3

Combine $2$ and $\frac{7}{7}$ .

$y = - \frac{5 x}{7} + \frac{10}{7} + \frac{2 \cdot 7}{7}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = - \frac{5 x}{7} + \frac{10 + 2 \cdot 7}{7}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $2$ by $7$ .

$y = - \frac{5 x}{7} + \frac{10 + 14}{7}$

Step 2.3.2.5.2

Add $10$ and $14$ .

$y = - \frac{5 x}{7} + \frac{24}{7}$

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{5}{7} x) + \frac{24}{7}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{5}{7} x + \frac{24}{7}$

Step 3