# Find the derivative?

##
y= x #sqrt(x^2-4#

y= x

##### 2 Answers

See the answer for the process on arriving to:

#### Explanation:

To find the derivative of

#y=xsqrt(x^2-4)#

we will (first) need to use the product rule. Recall that the product rule states that the derivative of the product of functions

The two functions being multiplied here are

#dy/dx=(d/dxx)sqrt(x^2-4)+x(d/dxsqrt(x^2-4))#

Note that

First, recall that

In all, we see that

#d/dx(x^2-4)^(1/2)=1/2(x^2-4)^(-1/2)(d/dx(x^2-4))#

The derivative of the inner function is

#d/dx(x^2-4)^(1/2)=1/2(x^2-4)^(-1/2)(2x)=x/sqrt(x^2-4)#

Returning to the whole function, substitute the two derivatives we've found in:

#dy/dx=(1)sqrt(x^2-4)+x(x/sqrt(x^2-4))#

And simplifying:

#dy/dx=sqrt(x^2-4)+x^2/sqrt(x^2-4)#

#dy/dx=((x^2-4)+x^2)/sqrt(x^2-4)#

#dy/dx=(2(x^2-2))/sqrt(x^2-4)#

#### Explanation:

We're attempting to find the **derivative of the product of two things**, so the **Product Rule** will help here.

First, I'll rewrite our equation in terms of functions. Thus, we have:

**NOTE**: **Chain Rule**- Inside function (

Product Rule:

Since we know both functions and their derivatives, we can plug in now. We get:

**NOTE:** We multiplied by the red expression to find a common denominator

After using the Product Rule and a good deal of algebraic manipulation, we were able to find the derivative of

Hope this helps!