# Is #f(x)=sin(pi/2-x)-cos(pi-x)# increasing or decreasing at #x=pi/3#?

##### 2 Answers

Decreasing.

#### Explanation:

We could use the chain rule to differentiate this. The chain rule, when specifically applied to the cosine and sine functions, is as follows:

When we apply this to the given function, we see that

Note that the derivative of each of these terms is

Now, to see if the function is increasing or decreasing, we must find the value of the derivative at

- If
#f'(pi/3)<0# , then#f(x)# is decreasing at#x=pi/3# . - If
#f'(pi/3)>0# , then#f(x)# is increasing at#x=pi/3# .

The value of the derivative at

We could continue and find the exact value, but we know that cosine is positive in the first quadrant, where

Since

Decreasing.

#### Explanation:

We could also expand these functions with the sine and cosine angle subtraction formulas.

#sin(a-b)=sin(a)cos(b)-cos(a)sin(b)# #cos(a-b)=cos(a)cos(b)+sin(a)sin(b)#

The function can then be written as

Simplify.

Differentiation is now simple if you know that

The value of the derivative at

Since this is negative, the function is decreasing at