Find the derivative?

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

2 Answers
Apr 25, 2018

See the answer for the process on arriving to:

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

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 #f# and #g# is given by #(fg)^'=f^'g+fg^'#.

The two functions being multiplied here are #x# and #sqrt(x^2-4)#, so we see that the derivative of #y# is given by

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

Note that #d/dxx=1#. In order to find #d/dxsqrt(x^2-4)#, we will need the chain rule.

First, recall that #sqrt(x^2-4)=(x^2-4)^(1/2)#. We differentiate this as we do #x^(1/2)#, with the power rule, bearing in mind that instead of just #x# we're working with the more complex function #x^2-4#. After applying the power rule, we use the chain rule, and multiply by the derivative of the inner function #x^2-4#.

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 #2x#, so we see that

#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)#

Apr 25, 2018

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

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:

#y=f(x)g(x)# where

#f(x)=x=>color(blue)(f'(x)=1)#

#g(x)=sqrt(x^2-4)=>color(lime)(g'(x)=x/(sqrt(x^2-4)))#

NOTE: #color(lime)(g'(x))# found via Chain Rule- Inside function (#x^2-4#), outside function (#x^(1/2)#)

Product Rule:

#f(x)g'(x)+f'(x)g(x)#

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

#dy/dx=x*color(lime)(x/(sqrt(x^2-4)))+color(blue)(1)*sqrt(x^2-4)#

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

#=(x^2)/((x^2-4)^(1/2))+(((x^2-4)^(1/2))/1*color(red)(((x^2-4)^(1/2))/((x^2-4)^(1/2))))#

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

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

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

#color(purple)(dy/dx=(2x^2-4)/(x^2-4)^(1/2))#

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

Hope this helps!