Need help finding the derivative of this function

[charmedbeauty]

Homework Statement

f(x)=x(x²-4)^1/2

Homework Equations

The Attempt at a Solution

so I started by, let,

u=x

u'=1

v=(x²-4)^1/2

v'=1/2(x²-4)^-1/2(2x)

so using the product rule

f'(x)= (x²-4)^1/2+x(1/2(x²-4)^-1/2(2x))

=(x²-4)^1/2+x²(x²-4)^-1/2

Now do I just differentiate the second term again I am not quiete understanding what needs to be done here??

 

[NewtonianAlch]

Where did the 4 come from?

 

[charmedbeauty]

Where did the 4 come from?

Sorry error in the question i'll edit now.

 

[Mentallic]

Homework Statement

f(x)=x(x²+1)^1/2
...
v=(x²-4)^1/2

Ok so you changed the question to $$f(x)=x\sqrt{x^2-4}$$ ?

f'(x)= √(x²-4)^1/2...

You have both the square root and the power to a half. Choose one or the other, but not both

Now do I just differentiate the second term again I am not quiete understanding what needs to be done here??

You have a function f(x) that is comprised of the product of two other functions g(x) and h(x), so $$f(x)=g(x)h(x)$$ where g(x)=x and $h(x)=\sqrt{x^2-4}$. Now, to take the derivative of f(x) we use the rule

$$f'(x)=g(x)h'(x)+g'(x)h(x)$$

Or, if we let g(x)=u and h(x)=v,

$$f'(x)=uv'+u'v$$

So once you've found the derivative of u and v, you plug u, v, u' and v' into the formula to obtain the derivative of f.

 

[charmedbeauty]

Ok so you changed the question to $$f(x)=x\sqrt{x^2-4}$$ ?



You have both the square root and the power to a half. Choose one or the other, but not both



You have a function f(x) that is comprised of the product of two other functions g(x) and h(x), so $$f(x)=g(x)h(x)$$ where g(x)=x and $h(x)=\sqrt{x^2-4}$. Now, to take the derivative of f(x) we use the rule

$$f'(x)=g(x)h'(x)+g'(x)h(x)$$

Or, if we let g(x)=u and h(x)=v,

$$f'(x)=uv'+u'v$$

So once you've found the derivative of u and v, you plug u, v, u' and v' into the formula to obtain the derivative of f.

But If you look at my working I have already done that? but I still get left with x²(x²-4)^-1/2+the first term.

but the answer says (2x²-4)/(x²-4)^1/2

for starters where does that fraction even come from?

and when I differentiate once I don't get the answer I get

=(x²-4)^1/2+x²(x²-4)^-1/2

 

[NewtonianAlch]

But If you look at my working I have already done that? but I still get left with x²(x²-4)^-1/2+the first term.

but the answer says (2x²-4)/(x²-4)^1/2

for starters where does that fraction even come from?

and when I differentiate once I don't get the answer I get

=(x²-4)^1/2+x²(x²-4)^-1/2

That second term should be x^2/(...) not x^2(...)

That answer is the same, they've just rationalised. Multiply through by the denominator of the fraction, and you will get the answer they gave. It's the same thing.

 

[charmedbeauty]

That second term should be x^2/(...) not x^2(...)

That answer is the same, they've just rationalised. Multiply through by the denominator of the fraction, and you will get the answer they gave. It's the same thing.

Ok Thanks!

 

[Mentallic]

But If you look at my working I have already done that?

Yes, I know. I was trying to help you get rid of your uncertainties as to what you need to do from there

 

[charmedbeauty]

Yes, I know. I was trying to help you get rid of your uncertainties as to what you need to do from there

OOOHHH thanks a bunch mentallic!