Calculus Examples

$\sin (x) \sin (x)$

Step 1

Raise

$\sin (x)$ to the power of

$1$ .

$\frac{d}{d x} [\sin^{1} (x) \sin (x)]$

Step 2

Raise

$\sin (x)$ to the power of

$1$ .

$\frac{d}{d x} [\sin^{1} (x) \sin^{1} (x)]$

Step 3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d x} [{\sin (x)}^{1 + 1}]$

Step 4

Add

$1$ and

$1$ .

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

Step 5

Differentiate using the chain rule, which states that

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

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

$f (x) = x^{2}$ and

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

Tap for more steps...

Step 5.1

To apply the Chain Rule, set

$u$ as

$\sin (x)$ .

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

Step 5.2

Differentiate using the Power Rule which states that

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

$n u^{n - 1}$ where

$n = 2$ .

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

Step 5.3

Replace all occurrences of

$u$ with

$\sin (x)$ .

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

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

Step 6

The derivative of

$\sin (x)$ with respect to

$x$ is

$\cos (x)$ .

$2 \sin (x) \cos (x)$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Reorder the factors of

$2 \sin (x) \cos (x)$ .

$2 \cos (x) \sin (x)$

Step 7.2

Reorder

$2 \cos (x)$ and

$\sin (x)$ .

$\sin (x) (2 \cos (x))$

Step 7.3

Reorder

$\sin (x)$ and

$2$ .

$2 \cdot \sin (x) \cos (x)$

Step 7.4

Apply the sine double-angle identity.

$\sin (2 x)$

$\sin (2 x)$

$\sin (x) \sin (x)$

(

(

)

)

|

|

[

[

]

]

√

√





≥

≥





∫

∫













7

7

8

8

9

9













≤

≤





°

°

θ

θ









4

4

5

5

6

6

/

/

^

^

×

×

>

>





π

π













1

1

2

2

3

3

-

-

+

+

÷

÷

<

<





∞

∞





!

!



,

,

0

0

.

.

%

%



=

=









$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$