# How do you differentiate #y=x^2+cos^-1x#?

##### 1 Answer

#### Explanation:

The derivative of

**DERIVATIVE OF**

Since you're expected to find the derivative of

The power rule states that the derivative of

So, for

**DERIVATIVE OF**

For this, we will need to do some manipulation. First, let:

#z=cos^-1x#

By the definition of the inverse trig functions (or inverse functions in general) this tells us that

#cos(z)=x#

We now should take the derivative of both sides (with respect to

#d/dxcos(z)=d/dxx#

#-sin(z)*(dz)/dx=1#

We then should solve for

#dz/dx=-1/sin(z)#

We can rewrite this in terms of our original function. Remember,

#d/dxcos^-1x=-1/sqrt(1-cos^2(z))#

And since

#d/dxcos^-1x=-1/sqrt(1-x^2)#

**PUTTING THEM TOGETHER**

We then see that:

#dy/dx=d/dxx^2+d/dxcos^-1x#

#dy/dx=2x+(-1/sqrt(1-x^2))#

#dy/dx=2x-1/sqrt(1-x^2)#