Question

Find the derivative?

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

Answer 1

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)`

Answer 2

`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!