Calculus Examples

$x \cos (y) = y$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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 $g (x) = y$ .

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Step 2.2.1

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

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

Step 2.2.2

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

$x (- \sin (u) \frac{d}{d x} [y]) + \cos (y) \frac{d}{d x} [x]$

Step 2.2.3

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

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

Step 2.3

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

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

Step 2.4

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

$x (- \sin (y) y') + \cos (y) \cdot 1$

Step 2.5

Simplify the expression.

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Step 2.5.1

Multiply $\cos (y)$ by $1$ .

$x (- \sin (y) y') + \cos (y)$

Step 2.5.2

Reorder terms.

$- x \sin (y) y' + \cos (y)$

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.

$- x \sin (y) y' + \cos (y) = y'$

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 $- x \sin (y) y' + \cos (y)$ .

$- x y' \sin (y) + \cos (y) = y'$

Step 5.2

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

$- x y' \sin (y) + \cos (y) - y' = 0$

Step 5.3

Subtract $\cos (y)$ from both sides of the equation.

$- x y' \sin (y) - y' = - \cos (y)$

Step 5.4

Factor $y'$ out of $- x y' \sin (y) - y'$ .

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Step 5.4.1

Factor $y'$ out of $- x y' \sin (y)$ .

$y' (- x \sin (y)) - y' = - \cos (y)$

Step 5.4.2

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

$y' (- x \sin (y)) + y' \cdot - 1 = - \cos (y)$

Step 5.4.3

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

$y' (- x \sin (y) - 1) = - \cos (y)$

Step 5.5

Divide each term in $y' (- x \sin (y) - 1) = - \cos (y)$ by $- x \sin (y) - 1$ and simplify.

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Step 5.5.1

Divide each term in $y' (- x \sin (y) - 1) = - \cos (y)$ by $- x \sin (y) - 1$ .

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $- x \sin (y) - 1$ .

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Step 5.5.2.1.1

Cancel the common factor.

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

Step 5.5.2.1.2

Divide $y'$ by $1$ .

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

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.

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

Step 5.5.3.2

Factor $- 1$ out of $- x \sin (y)$ .

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

Step 5.5.3.3

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

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

Step 5.5.3.4

Factor $- 1$ out of $- (x \sin (y)) - 1 (1)$ .

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

Step 5.5.3.5

Simplify the expression.

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Step 5.5.3.5.1

Rewrite $- (x \sin (y) + 1)$ as $- 1 (x \sin (y) + 1)$ .

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

Step 5.5.3.5.2

Move the negative in front of the fraction.

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

Step 5.5.3.5.3

Multiply $- 1$ by $- 1$ .

$y' = 1 \frac{\cos (y)}{x \sin (y) + 1}$

Step 5.5.3.5.4

Multiply $\frac{\cos (y)}{x \sin (y) + 1}$ by $1$ .

$y' = \frac{\cos (y)}{x \sin (y) + 1}$

Step 6

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

$\frac{d y}{d x} = \frac{\cos (y)}{x \sin (y) + 1}$