Calculus Examples

$y^{3} x^{5} = e^{3 x}$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = y^{3}$ and $g (y) = x^{5}$ .

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

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{5}$ and $g (y) = x$ .

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Step 2.2.1

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

$y^{3} (\frac{d}{d u} [u^{5}] \frac{d}{d y} [x]) + x^{5} \frac{d}{d y} [y^{3}]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$y^{3} (5 u^{4} \frac{d}{d y} [x]) + x^{5} \frac{d}{d y} [y^{3}]$

Step 2.2.3

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

$y^{3} (5 x^{4} \frac{d}{d y} [x]) + x^{5} \frac{d}{d y} [y^{3}]$

Step 2.3

Move $5$ to the left of $y^{3}$ .

$5 \cdot y^{3} x^{4} \frac{d}{d y} [x] + x^{5} \frac{d}{d y} [y^{3}]$

Step 2.4

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

$5 y^{3} x^{4} x' + x^{5} \frac{d}{d y} [y^{3}]$

Step 2.5

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

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

Step 2.6

Reorder.

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Step 2.6.1

Move $3$ to the left of $x^{5}$ .

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

Step 2.6.2

Reorder terms.

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

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

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Step 3.1.1

To apply the Chain Rule, set $u$ as $3 x$ .

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

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u$ with $3 x$ .

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

Step 3.2

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

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

Step 3.3

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

$e^{3 x} (3 x')$

Step 3.4

Reorder the factors of $e^{3 x} (3 x')$ .

$3 e^{3 x} x'$

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

Reorder factors in $3 e^{3 x} x'$ .

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

Step 5.2

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

$5 x^{4} y^{3} x' + 3 x^{5} y^{2} - 3 x' e^{3 x} = 0$

Step 5.3

Subtract $3 x^{5} y^{2}$ from both sides of the equation.

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

Step 5.4

Factor $x'$ out of $5 x^{4} y^{3} x' - 3 x' e^{3 x}$ .

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Step 5.4.1

Factor $x'$ out of $5 x^{4} y^{3} x'$ .

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

Step 5.4.2

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

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

Step 5.4.3

Factor $x'$ out of $x' (5 x^{4} y^{3}) + x' (- 3 e^{3 x})$ .

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $5 x^{4} y^{3} - 3 e^{3 x}$ .

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Step 5.5.2.1.1

Cancel the common factor.

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

Step 5.5.2.1.2

Divide $x'$ by $1$ .

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

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Move the negative in front of the fraction.

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

Step 6

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

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