29) Find the derivative of the function.
F(X)=2x^2(3-4x)^2
This is what I have so far. I am usually pretty good at simplifying but I am missing something here.
f'(x)=2x^2(4)(3-4x)^3(-4) + (3-4x)^4(4x)
Here is where I go wrong. (Answer from book)
=4x(3-4x)^3(-12+3)
Thanks for your help.
Should be =4x(3-4x)^3(-12x+3)
F(X)=2x^2(3-4x)^2
This is what I have so far. I am usually pretty good at simplifying but I am missing something here.
f'(x)=2x^2(4)(3-4x)^3(-4) + (3-4x)^4(4x)
Here is where I go wrong. (Answer from book)
=4x(3-4x)^3(-12+3)
=================================
I get
F(X)=2x^2(3-4x)^2
f' = 2x^2 (2)(3-4x)(-4) +(3-4x)^2(4x)
= -16x^2(3-4x)+4x(3-4x)^2
= 4x(3-4x)(3-4x) -16x^2(3-4x)
= 4x(3-4x)(3-4x) -4x(3-4x)(4x)
= 4x(3-4x)(3-4x-4x)
=4x(3-4x)(3-8x)
F(X)=2x^2(3-4x)^2
another way
f(x) = 2x^2(9 - 24x +16 x^2)
= 18 x^2 - 48 x^3 +32 x^4
take the derivative of that
= 36 x - 144 x^2 + 128 x^3
= 4x ( 9 - 36 x + 32 x^2)
= 4 x (3-4x)(3-8x)
Unless you have a typo, I have it right and the book has it wrong.
The final answer in the book is
(-12x)(4x-1)(3-4x)^3
This book has had several errors so far.
In google type:
wolfram alpha
When you see list of results click on:
Wolfram Alpha:Comutation Knowledge Engine
When page be open in rectangle type:
2x^2(3-4x)^2
and click option =
After few seconds you will see everything about that function.
Then click option
Derivative: Show steps
To find the derivative of the function F(x) = 2x^2(3 - 4x)^2, we can use the product rule. The product rule states that the derivative of a product of two functions is equal to the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function.
Let's break down the steps involved in finding the derivative:
Step 1: Apply the product rule
f'(x) = (2x^2)'(3 - 4x)^2 + 2x^2((3 - 4x)^2)'
Step 2: Differentiate the first term
To find the derivative of 2x^2, we apply the power rule for differentiation. The power rule states that for any real number n, the derivative of x^n with respect to x is nx^(n-1).
(2x^2)' = 2 * 2x^(2-1) = 4x
Step 3: Differentiate the second term
To differentiate (3 - 4x)^2, we can use the chain rule. The chain rule states that if we have a function g(h(x)), then the derivative of g with respect to x is equal to the derivative of g with respect to h, multiplied by the derivative of h with respect to x.
(3 - 4x)^2' = 2(3 - 4x)'(3 - 4x) = 2(-4)(3 - 4x) = -8(3 - 4x)
Step 4: Combine the derivatives and simplify
Substitute the derivatives back into the derivative expression from step 1:
f'(x) = 4x * (3 - 4x)^2 + 2x^2 * (-8)(3 - 4x)
Now, to simplify further, distribute the terms and combine like terms:
f'(x) = 4x * (9 - 24x + 16x^2) - 16x^2 * (3 - 4x)
Expanding and combining like terms:
f'(x) = 36x - 96x^2 + 64x^3 - 48x^2 + 64x^3
Simplifying further:
f'(x) = 128x^3 - 144x^2 + 36x
So, the derivative of the function F(x) = 2x^2(3 - 4x)^2 is f'(x) = 128x^3 - 144x^2 + 36x.