Calculus Examples

sin (x y)

$\sin (x y)$

Step 1

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = \sin (x)$ and

$g (x) = x y$ .

Tap for more steps...

Step 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 1.2

The derivative of

$\sin (u)$ with respect to

$u$ is

$\cos (u)$ .

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

Step 1.3

Replace all occurrences of

$u$ with

$x y$ .

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

Since

$y$ is constant with respect to

$x$ , the derivative of

$x y$ with respect to

$x$ is

$y \frac{d}{d x} [x]$ .

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

Step 2.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$\cos (x y) (y \cdot 1)$

Step 2.3

Simplify the expression.

Tap for more steps...

Step 2.3.1

Multiply

$y$ by

$1$ .

$\cos (x y) y$

Step 2.3.2

Reorder the factors of

$\cos (x y) y$ .

$y \cos (x y)$

$\sin (x y)$

(

)

[

]

√



≥



∫













≤

















∞













