Calculus Examples

$y = \sin (x + y)$

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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 $g (x) = x + y$ .

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

Differentiate.

Tap for more steps...

Step 3.2.1

By the Sum Rule, the derivative of $x + y$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [y]$ .

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

Step 3.2.2

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

$\cos (x + y) (1 + \frac{d}{d x} [y])$

Step 3.3

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

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

Step 4

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

$y' = \cos (x + y) (1 + y')$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Simplify the right side.

Tap for more steps...

Step 5.1.1

Simplify $\cos (x + y) (1 + y')$ .

Tap for more steps...

Step 5.1.1.1

Apply the distributive property.

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

Step 5.1.1.2

Simplify the expression.

Tap for more steps...

Step 5.1.1.2.1

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

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

Step 5.1.1.2.2

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

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

Step 5.2

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

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

Step 5.3

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

Tap for more steps...

Step 5.3.1

Factor $y'$ out of ${y'}^{1}$ .

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.4

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

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

Step 5.5

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

Tap for more steps...

Step 5.5.1

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

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

Step 5.5.2

Simplify the left side.

Tap for more steps...

Step 5.5.2.1

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

Tap for more steps...

Step 5.5.2.1.1

Cancel the common factor.

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

Step 5.5.2.1.2

Divide $y'$ by $1$ .

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

Step 6

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

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