Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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Step 2.1.1

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

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

Step 2.1.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

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

Step 2.1.3

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

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

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 $g (x) = y$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Reorder terms.

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

Step 3

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

$1$

Step 4

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

$x^{2} \cos (x^{2} y) y' + 2 x y \cos (x^{2} y) = 1$

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

$x^{2} y' \cos (x^{2} y) + 2 x y \cos (x^{2} y) = 1$

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

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

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Rewrite the expression.

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

Step 5.3.2.2

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

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Divide $y'$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Separate fractions.

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

Step 5.3.3.1.2

Convert from $\frac{1}{\cos (x^{2} y)}$ to $\sec (x^{2} y)$ .

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

Step 5.3.3.1.3

Combine $\frac{1}{x^{2}}$ and $\sec (x^{2} y)$ .

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

Step 5.3.3.1.4

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

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Step 5.3.3.1.4.1

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

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

Step 5.3.3.1.4.2

Cancel the common factors.

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Step 5.3.3.1.4.2.1

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

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

Step 5.3.3.1.4.2.2

Cancel the common factor.

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

Step 5.3.3.1.4.2.3

Rewrite the expression.

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

Step 5.3.3.1.5

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

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Step 5.3.3.1.5.1

Cancel the common factor.

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

Step 5.3.3.1.5.2

Rewrite the expression.

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

Step 5.3.3.1.6

Move the negative in front of the fraction.

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

Step 6

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

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