Calculus Examples

$y^{3} e^{x} + x = y^{6} x$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y^{3} e^{x} + x) = \frac{d}{d x} (y^{6} x)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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 $f (x) = y^{3}$ and $g (x) = e^{x}$ .

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

Step 2.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

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

Step 2.2.3

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 2.2.3.1

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

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

Step 2.2.3.2

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

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

Step 2.2.3.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Move $3$ to the left of $e^{x}$ .

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

Step 2.3

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

$y^{3} e^{x} + 3 e^{x} y^{2} y' + 1$

Step 2.4

Simplify.

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Step 2.4.1

Reorder terms.

$e^{x} y^{3} + 3 e^{x} y^{2} y' + 1$

Step 2.4.2

Reorder factors in $e^{x} y^{3} + 3 e^{x} y^{2} y' + 1$ .

$y^{3} e^{x} + 3 y^{2} e^{x} y' + 1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = y^{6}$ and $g (x) = x$ .

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

Step 3.2

Differentiate using the Power Rule.

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Step 3.2.1

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

$y^{6} \cdot 1 + x \frac{d}{d x} [y^{6}]$

Step 3.2.2

Multiply $y^{6}$ by $1$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

Move $6$ to the left of $x$ .

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

Step 3.5

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

$y^{6} + 6 x y^{5} y'$

Step 3.6

Reorder terms.

$y^{6} + 6 y^{5} x y'$

Step 4

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

$y^{3} e^{x} + 3 y^{2} e^{x} y' + 1 = y^{6} + 6 y^{5} x y'$

Step 5

Solve for $y'$ .

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Step 5.1

Reorder factors in $y^{3} e^{x} + 3 y^{2} e^{x} y' + 1$ .

$y^{3} e^{x} + 3 y^{2} y' e^{x} + 1 = y^{6} + 6 y^{5} x y'$

Step 5.2

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

$y^{3} e^{x} + 3 y^{2} y' e^{x} + 1 - 6 y^{5} x y' = y^{6}$

Step 5.3

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

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Step 5.3.1

Subtract $y^{3} e^{x}$ from both sides of the equation.

$3 y^{2} y' e^{x} + 1 - 6 y^{5} x y' = y^{6} - y^{3} e^{x}$

Step 5.3.2

Subtract $1$ from both sides of the equation.

$3 y^{2} y' e^{x} - 6 y^{5} x y' = y^{6} - y^{3} e^{x} - 1$

Step 5.4

Factor $3 y^{2} y'$ out of $3 y^{2} y' e^{x} - 6 y^{5} x y'$ .

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Step 5.4.1

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

$3 y^{2} y' e^{x} - 6 y^{5} x y' = y^{6} - y^{3} e^{x} - 1$

Step 5.4.2

Factor $3 y^{2} y'$ out of $- 6 y^{5} x y'$ .

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

Step 5.4.3

Factor $3 y^{2} y'$ out of $3 y^{2} y' e^{x} + 3 y^{2} y' (- 2 y^{3} x)$ .

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $3$ .

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Step 5.5.2.1.1

Cancel the common factor.

Step 5.5.2.1.2

Rewrite the expression.

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

Step 5.5.2.2

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

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Step 5.5.2.2.1

Cancel the common factor.

Step 5.5.2.2.2

Rewrite the expression.

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

Step 5.5.2.3

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

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Step 5.5.2.3.1

Cancel the common factor.

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

Step 5.5.2.3.2

Divide $y'$ by $1$ .

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

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Cancel the common factor of $y^{6}$ and $y^{2}$ .

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Step 5.5.3.1.1.1

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

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

Step 5.5.3.1.1.2

Cancel the common factors.

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Step 5.5.3.1.1.2.1

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

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

Step 5.5.3.1.1.2.2

Cancel the common factor.

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

Step 5.5.3.1.1.2.3

Rewrite the expression.

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

Step 5.5.3.1.2

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

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Step 5.5.3.1.2.1

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

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

Step 5.5.3.1.2.2

Cancel the common factors.

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Step 5.5.3.1.2.2.1

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

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

Step 5.5.3.1.2.2.2

Cancel the common factor.

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

Step 5.5.3.1.2.2.3

Rewrite the expression.

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

Step 5.5.3.1.3

Move the negative in front of the fraction.

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

Step 5.5.3.1.4

Move the negative in front of the fraction.

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

Step 6

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

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