Find the derivative of the following function showing your work and fully simplifying your answer. Show steps please! Thank you!!!!!!
f(x)=(x+3)^3(x-8)^5
I got up to here.
f'(x)= 3(x+3)^2(x8)^5(d/dx(x))+5(x+3)^3(x-8)^4
d/dx(x) = dx/dx = 1
So, how come you included it in d/dx (x+3) but not in d/dx (x-8)?
Your work so far is correct.
3(x+3)^2(x-8)^5(d/dx(x))+5(x+3)^3(x-8)^4
3(x+3)^2(x-8)^5 + 5(x+3)^3(x-8)^4
Now factor out some stuff:
(x+3)^2 (x-8)^4 (3(x-8)+5(x+3))
(x+3)^2 (x-8)^4 (8x-9)
To find the derivative of the function f(x) = (x+3)^3(x-8)^5, we can use the product rule and chain rule of differentiation. Here are the steps to simplify the answer:
Step 1: Expand the function
f(x) = (x+3)^3(x-8)^5
= (x+3)(x+3)(x+3)(x-8)(x-8)(x-8)
Step 2: Find the derivative using the product rule
Let's differentiate each factor separately:
a) (x+3)(x+3)(x+3):
Using the chain rule, we differentiate the first factor (x+3) with respect to x:
d/dx (x+3) = 1
For the other two factors, we keep them as they are.
Therefore, the derivative of (x+3)(x+3)(x+3) is:
3(x+3)(x+3)
b) (x-8)(x-8)(x-8):
Using the chain rule, we differentiate the first factor (x-8) with respect to x:
d/dx (x-8) = 1
For the other two factors, we keep them as they are.
Therefore, the derivative of (x-8)(x-8)(x-8) is:
3(x-8)(x-8)
Step 3: Apply the product rule
The derivative of the function f(x) = (x+3)^3(x-8)^5 is obtained by multiplying the derivative of the first factor (Step 2a) with the other two factors, and the derivative of the second factor (Step 2b) with the other two factors, and then summing them up.
Therefore, the derivative of f(x) is:
[3(x+3)(x+3)](x-8)(x-8)(x-8) + (x+3)(x+3)[3(x-8)(x-8)]
Simplifying this expression by distributing and combining like terms, we get the fully simplified answer.