Calculus Examples

$y \sin (x) = x \sin (y)$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

Tap for more steps...

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 $f (x) = y$ and $g (x) = \sin (x)$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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 $g (x) = \sin (y)$ .

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 4

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

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

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Simplify the left side.

Tap for more steps...

Step 5.1.1

Reorder factors in $y \cos (x) + \sin (x) y'$ .

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

Step 5.2

Simplify the right side.

Tap for more steps...

Step 5.2.1

Reorder factors in $x \cos (y) y' + \sin (y)$ .

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

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

Tap for more steps...

Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

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

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

Step 5.6

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

Tap for more steps...

Step 5.6.1

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

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

Step 5.6.2

Simplify the left side.

Tap for more steps...

Step 5.6.2.1

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

Tap for more steps...

Step 5.6.2.1.1

Cancel the common factor.

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

Step 5.6.2.1.2

Divide $y'$ by $1$ .

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

Step 5.6.3

Simplify the right side.

Tap for more steps...

Step 5.6.3.1

Move the negative in front of the fraction.

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

Step 5.6.3.2

Combine the numerators over the common denominator.

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

Step 6

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

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