Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

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

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

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 $y$ .

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

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 $n = 2$ .

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

Step 2.3.3

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

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

Step 2.4

Move $2$ to the left of $x$ .

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

Step 2.5

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

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

Step 2.6

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

$- \sin (x y^{2}) (2 x y y' + y^{2} \cdot 1)$

Step 2.7

Multiply $y^{2}$ by $1$ .

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

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

$- \sin (x y^{2}) (2 x y y') - \sin (x y^{2}) y^{2}$

Step 2.8.2

Multiply $2$ by $- 1$ .

$- 2 \sin (x y^{2}) (x y y') - \sin (x y^{2}) y^{2}$

Step 2.8.3

Reorder terms.

$- 2 x y \sin (x y^{2}) y' - y^{2} \sin (x y^{2})$

Step 3

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

$y'$

Step 4

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

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

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 $- 2 x y \sin (x y^{2}) y' - y^{2} \sin (x y^{2})$ .

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

Step 5.2

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

$- 2 x y y' \sin (x y^{2}) - y^{2} \sin (x y^{2}) - y' = 0$

Step 5.3

Add $y^{2} \sin (x y^{2})$ to both sides of the equation.

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

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

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

Step 5.4.3

Factor $y'$ out of $y' (- 2 x y \sin (x y^{2})) + y' \cdot - 1$ .

$y' (- 2 x y \sin (x y^{2}) - 1) = y^{2} \sin (x y^{2})$

Step 5.5

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

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Step 5.5.1

Divide each term in $y' (- 2 x y \sin (x y^{2}) - 1) = y^{2} \sin (x y^{2})$ by $- 2 x y \sin (x y^{2}) - 1$ .

$\frac{y' (- 2 x y \sin (x y^{2}) - 1)}{- 2 x y \sin (x y^{2}) - 1} = \frac{y^{2} \sin (x y^{2})}{- 2 x y \sin (x y^{2}) - 1}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $- 2 x y \sin (x y^{2}) - 1$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{y' (- 2 x y \sin (x y^{2}) - 1)}{- 2 x y \sin (x y^{2}) - 1} = \frac{y^{2} \sin (x y^{2})}{- 2 x y \sin (x y^{2}) - 1}$

Step 5.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{y^{2} \sin (x y^{2})}{- 2 x y \sin (x y^{2}) - 1}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Factor $- 1$ out of $- 2 x y \sin (x y^{2})$ .

$y' = \frac{y^{2} \sin (x y^{2})}{- (2 x y \sin (x y^{2})) - 1}$

Step 5.5.3.2

Rewrite $- 1$ as $- 1 (1)$ .

$y' = \frac{y^{2} \sin (x y^{2})}{- (2 x y \sin (x y^{2})) - 1 (1)}$

Step 5.5.3.3

Factor $- 1$ out of $- (2 x y \sin (x y^{2})) - 1 (1)$ .

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

Step 5.5.3.4

Rewrite negatives.

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Step 5.5.3.4.1

Rewrite $- (2 x y \sin (x y^{2}) + 1)$ as $- 1 (2 x y \sin (x y^{2}) + 1)$ .

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

Step 5.5.3.4.2

Move the negative in front of the fraction.

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

Step 6

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

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