Derivative

Question

Find the derivative of f(x)=xsquared(2x-3)to power of 4

 

sorry i couldn't write the powers out on my phone. Can someone please explain the question?

Answer 1

Hi Iona,

 

So f(x) = x² (2x-3)⁴

 

First, let's decide what differentiation rules we will need to use (product rule, quotient rule, etc...). It looks to me that f(x) involves two "sub-functions" of x being multiplied together, x² and (2x-3)⁴ , so I know we will need to use the product rule. Furthermore, the (2x-3)⁴ "sub-function" looks like a "function inside of a function", so taking its derivative will involve using the chain rule. I'm going to call these two "sub-functions" g(x) and h(x), just so we don't confuse them with the overall f(x).

 

    g(x) = x²

 

    h(x) = (2x-3)⁴

 

The product rule says that the derivative of two functions multiplied together, g(x)*h(x) is:

 

    g'(x)*h(x) + h'(x)*g(x)

 

Now we just need to identify all four of these parts, and put them into that expression. We already know g(x) and h(x), we just need to find each of their derivatives.

 

For g(x), we use the power rule:

   

    g'(x) = 2x

 

For h(x), we use the chain rule - find the derivative of the outside function, with the inside function plugged in as the input, then multiply by the derivative of the inside function itself:

 

    h'(x) = 4(2x-3)³ * 2 = 8(2x-3)³

 

Now we plug these four functions into the product rule expression, and simplify if possible:

 

    g'(x)*h(x) + h'(x)*g(x)

 

    (2x)*(2x-3)⁴ + (8(2x-3)³)*(x²)

 

I hope this helps!

 

Andy

 

    

Answer 2

f'(x) = derivative(x²).(2x-3)⁴  + x². derivative( (2x-3)⁴) = 2x(2x-3)⁴ + x²4.(2x-3)³.2

        = x(2x-3)³ [ 2x(2x-3) +8x^2] =x(2x-3)³(12x² -12x) = 12(x-1)x²(2x-3)³