Calculus Examples

$2 y^{3} + 4 x^{2} - y = x^{6}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (2 y^{3} + 4 x^{2} - y) = \frac{d}{d y} (x^{6})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $2 y^{3} + 4 x^{2} - y$ with respect to $y$ is $\frac{d}{d y} [2 y^{3}] + \frac{d}{d y} [4 x^{2}] + \frac{d}{d y} [- y]$ .

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

Step 2.2

Evaluate $\frac{d}{d y} [2 y^{3}]$ .

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Step 2.2.1

Since $2$ is constant with respect to $y$ , the derivative of $2 y^{3}$ with respect to $y$ is $2 \frac{d}{d y} [y^{3}]$ .

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $3$ by $2$ .

$6 y^{2} + \frac{d}{d y} [4 x^{2}] + \frac{d}{d y} [- y]$

Step 2.3

Evaluate $\frac{d}{d y} [4 x^{2}]$ .

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Step 2.3.1

Since $4$ is constant with respect to $y$ , the derivative of $4 x^{2}$ with respect to $y$ is $4 \frac{d}{d y} [x^{2}]$ .

$6 y^{2} + 4 \frac{d}{d y} [x^{2}] + \frac{d}{d y} [- y]$

Step 2.3.2

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

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Step 2.3.2.1

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

$6 y^{2} + 4 (\frac{d}{d u} [u^{2}] \frac{d}{d y} [x]) + \frac{d}{d y} [- y]$

Step 2.3.2.2

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

$6 y^{2} + 4 (2 u \frac{d}{d y} [x]) + \frac{d}{d y} [- y]$

Step 2.3.2.3

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

$6 y^{2} + 4 (2 x \frac{d}{d y} [x]) + \frac{d}{d y} [- y]$

Step 2.3.3

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

$6 y^{2} + 4 (2 x x') + \frac{d}{d y} [- y]$

Step 2.3.4

Multiply $2$ by $4$ .

$6 y^{2} + 8 x x' + \frac{d}{d y} [- y]$

Step 2.4

Evaluate $\frac{d}{d y} [- y]$ .

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Step 2.4.1

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

$6 y^{2} + 8 x x' - \frac{d}{d y} [y]$

Step 2.4.2

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

$6 y^{2} + 8 x x' - 1 \cdot 1$

Step 2.4.3

Multiply $- 1$ by $1$ .

$6 y^{2} + 8 x x' - 1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{6}$ and $g (y) = x$ .

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Step 3.1.1

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

$\frac{d}{d u} [u^{6}] \frac{d}{d y} [x]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 6$ .

$6 u^{5} \frac{d}{d y} [x]$

Step 3.1.3

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

$6 x^{5} \frac{d}{d y} [x]$

Step 3.2

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

$6 x^{5} x'$

Step 4

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

$6 y^{2} + 8 x x' - 1 = 6 x^{5} x'$

Step 5

Solve for $x'$ .

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Step 5.1

Subtract $6 x^{5} x'$ from both sides of the equation.

$6 y^{2} + 8 x x' - 1 - 6 x^{5} x' = 0$

Step 5.2

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

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Step 5.2.1

Subtract $6 y^{2}$ from both sides of the equation.

$8 x x' - 1 - 6 x^{5} x' = - 6 y^{2}$

Step 5.2.2

Add $1$ to both sides of the equation.

$8 x x' - 6 x^{5} x' = - 6 y^{2} + 1$

Step 5.3

Factor $2 x x'$ out of $8 x x' - 6 x^{5} x'$ .

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Step 5.3.1

Factor $2 x x'$ out of $8 x x'$ .

$2 x x' \cdot 4 - 6 x^{5} x' = - 6 y^{2} + 1$

Step 5.3.2

Factor $2 x x'$ out of $- 6 x^{5} x'$ .

$2 x x' \cdot 4 + 2 x x' (- 3 x^{4}) = - 6 y^{2} + 1$

Step 5.3.3

Factor $2 x x'$ out of $2 x x' \cdot 4 + 2 x x' (- 3 x^{4})$ .

$2 x x' (4 - 3 x^{4}) = - 6 y^{2} + 1$

Step 5.4

Divide each term in $2 x x' (4 - 3 x^{4}) = - 6 y^{2} + 1$ by $2 x (4 - 3 x^{4})$ and simplify.

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Step 5.4.1

Divide each term in $2 x x' (4 - 3 x^{4}) = - 6 y^{2} + 1$ by $2 x (4 - 3 x^{4})$ .

$\frac{2 x x' (4 - 3 x^{4})}{2 x (4 - 3 x^{4})} = \frac{- 6 y^{2}}{2 x (4 - 3 x^{4})} + \frac{1}{2 x (4 - 3 x^{4})}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $2$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{2 x x' (4 - 3 x^{4})}{2 x (4 - 3 x^{4})} = \frac{- 6 y^{2}}{2 x (4 - 3 x^{4})} + \frac{1}{2 x (4 - 3 x^{4})}$

Step 5.4.2.1.2

Rewrite the expression.

$\frac{x x' (4 - 3 x^{4})}{x (4 - 3 x^{4})} = \frac{- 6 y^{2}}{2 x (4 - 3 x^{4})} + \frac{1}{2 x (4 - 3 x^{4})}$

Step 5.4.2.2

Cancel the common factor of $x$ .

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Step 5.4.2.2.1

Cancel the common factor.

$\frac{x x' (4 - 3 x^{4})}{x (4 - 3 x^{4})} = \frac{- 6 y^{2}}{2 x (4 - 3 x^{4})} + \frac{1}{2 x (4 - 3 x^{4})}$

Step 5.4.2.2.2

Rewrite the expression.

$\frac{x' (4 - 3 x^{4})}{4 - 3 x^{4}} = \frac{- 6 y^{2}}{2 x (4 - 3 x^{4})} + \frac{1}{2 x (4 - 3 x^{4})}$

Step 5.4.2.3

Cancel the common factor of $4 - 3 x^{4}$ .

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Step 5.4.2.3.1

Cancel the common factor.

$\frac{x' (4 - 3 x^{4})}{4 - 3 x^{4}} = \frac{- 6 y^{2}}{2 x (4 - 3 x^{4})} + \frac{1}{2 x (4 - 3 x^{4})}$

Step 5.4.2.3.2

Divide $x'$ by $1$ .

$x' = \frac{- 6 y^{2}}{2 x (4 - 3 x^{4})} + \frac{1}{2 x (4 - 3 x^{4})}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

Cancel the common factor of $- 6$ and $2$ .

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Step 5.4.3.1.1.1

Factor $2$ out of $- 6 y^{2}$ .

$x' = \frac{2 (- 3 y^{2})}{2 x (4 - 3 x^{4})} + \frac{1}{2 x (4 - 3 x^{4})}$

Step 5.4.3.1.1.2

Cancel the common factors.

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Step 5.4.3.1.1.2.1

Factor $2$ out of $2 x (4 - 3 x^{4})$ .

$x' = \frac{2 (- 3 y^{2})}{2 (x (4 - 3 x^{4}))} + \frac{1}{2 x (4 - 3 x^{4})}$

Step 5.4.3.1.1.2.2

Cancel the common factor.

$x' = \frac{2 (- 3 y^{2})}{2 (x (4 - 3 x^{4}))} + \frac{1}{2 x (4 - 3 x^{4})}$

Step 5.4.3.1.1.2.3

Rewrite the expression.

$x' = \frac{- 3 y^{2}}{x (4 - 3 x^{4})} + \frac{1}{2 x (4 - 3 x^{4})}$

Step 5.4.3.1.2

Move the negative in front of the fraction.

$x' = - \frac{3 y^{2}}{x (4 - 3 x^{4})} + \frac{1}{2 x (4 - 3 x^{4})}$

Step 6

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

$\frac{d x}{d y} = - \frac{3 y^{2}}{x (4 - 3 x^{4})} + \frac{1}{2 x (4 - 3 x^{4})}$