Calculus Examples

$\sin (x^{2} y^{2}) = 3 y$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $x^{2} y^{2}$ .

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [x^{2} y^{2}]$

Step 2.1.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$\cos (u_{1}) \frac{d}{d x} [x^{2} y^{2}]$

Step 2.1.3

Replace all occurrences of $u_{1}$ with $x^{2} y^{2}$ .

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

Step 2.2

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 $f (x) = x^{2}$ and $g (x) = y^{2}$ .

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

Step 2.3

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

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Step 2.3.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

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

Step 2.3.2

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

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

Step 2.3.3

Replace all occurrences of $u_{2}$ with $y$ .

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

Step 2.4

Move $2$ to the left of $x^{2}$ .

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

Step 2.5

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

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

Step 2.6

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

$\cos (x^{2} y^{2}) (2 x^{2} y y' + y^{2} (2 x))$

Step 2.7

Move $2$ to the left of $y^{2}$ .

$\cos (x^{2} y^{2}) (2 x^{2} y y' + 2 \cdot y^{2} x)$

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

$\cos (x^{2} y^{2}) (2 x^{2} y y') + \cos (x^{2} y^{2}) (2 y^{2} x)$

Step 2.8.2

Reorder terms.

$2 x^{2} y \cos (x^{2} y^{2}) y' + 2 y^{2} x \cos (x^{2} y^{2})$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

$3 y'$

Step 4

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

$2 x^{2} y \cos (x^{2} y^{2}) y' + 2 y^{2} x \cos (x^{2} y^{2}) = 3 y'$

Step 5

Solve for $y'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

Reorder factors in $2 x^{2} y \cos (x^{2} y^{2}) y' + 2 y^{2} x \cos (x^{2} y^{2})$ .

$2 x^{2} y y' \cos (x^{2} y^{2}) + 2 y^{2} x \cos (x^{2} y^{2}) = 3 y'$

Step 5.2

Subtract $3 y'$ from both sides of the equation.

$2 x^{2} y y' \cos (x^{2} y^{2}) + 2 y^{2} x \cos (x^{2} y^{2}) - 3 y' = 0$

Step 5.3

Subtract $2 y^{2} x \cos (x^{2} y^{2})$ from both sides of the equation.

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

Step 5.4

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

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Step 5.4.1

Factor $y'$ out of $2 x^{2} y y' \cos (x^{2} y^{2})$ .

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

Step 5.4.2

Factor $y'$ out of $- 3 y'$ .

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

Step 5.4.3

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

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $2 x^{2} y \cos (x^{2} y^{2}) - 3$ .

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Step 5.5.2.1.1

Cancel the common factor.

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

Step 5.5.2.1.2

Divide $y'$ by $1$ .

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

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.

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

Step 6

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

$\frac{d y}{d x} = - \frac{2 y^{2} x \cos (x^{2} y^{2})}{2 x^{2} y \cos (x^{2} y^{2}) - 3}$