Find the derivative of the equation f(x)?

Product Rule

a = x b= (4-x)^3

a '= 1 b'= 3(4-x)^2(-1)

b'= -3(4-x)^2

f'(x)= ab'+ba'

f'(x)= (x) (-3(4-x)^2) + (4-x)^3 (1)

The a and the b show the different segments of the problem. For this problem i used both the chain rule and the product rule. The chain rules was used to find the derivative of B. And the product rule was used to figure out the derivative of the problem as a whole.

Given the function: f(x) = x * (4 - x)^(3)

Take the derivative to both sides of the function with respect to x.

d/dx[f(x)] = d/dx[ x * (4 - x)^(3) ]

Use the product rule of differentiation: d/dx[uv]= uv'+vu'

u = x

u ' = 1

v = (4 - x)^(3)

v ' = 3 (4 - x )^(2) * ( - 1) = - 3(4 - x)^(2) ----> Used Chain Rule here

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f ' (x) = [x] * [ 3 (4 - x) ^(2) * (- 1) ] + [ (4 - x)^(3) ] * [(1)]

Multiply using the distributive property.

f ' (x) = [x] * [ - 3(4 - x)^(2) ] + [(4 - x)^(3)]

Multiply using the distributive property.

f ' (x) = - 3x (4 - x)^(2) + (4 - x)^(3)

Factor out a (4 - x)^(2) here.

f ' (x) = (4 - x)^(2) * [ -3x + (4 - x)]

Combine like terms inside the brackets.

f ' (x) = (4 - x)^(2) * [- 4x + 4 ] ---> ANSWER

Start with the product rule.

f'(x) = dx/dx * (4-x)^3 + x * d/dx (4-x)^3

Then use the chain rule

d/dx (4-x)^3 = 3(4-x)^2 * d/dx (4-x)