Find ther derivative of the functions.
a. f(x)=(4x-x^2)^100
b. f(x)=(1+2x+x^3)^1/4
apply the chain rule. If y = f(u), dy/dx = dy/du * du/dx
a. u = 4x-x^2, y = u^100
b. u = 1+2x+x^3, y = u^1/4
Got it?
To find the derivatives of these functions, we can use the chain rule. The chain rule states that the derivative of a composite function is given by the derivative of the outer function multiplied by the derivative of the inner function.
a. To find the derivative of f(x) = (4x - x^2)^100, we can rewrite it as g(x)^100, where g(x) = 4x - x^2.
Step 1: Find the derivative of g(x)
g'(x) = d/dx (4x - x^2)
= 4 - 2x
Step 2: Apply the chain rule to find the derivative of f(x)
f'(x) = d/dx [(4x - x^2)^100]
= 100(4x - x^2)^99 * d/dx (4x - x^2)
= 100(4x - x^2)^99 * (4 - 2x)
Therefore, the derivative of f(x) = (4x - x^2)^100 is f'(x) = 100(4x - x^2)^99 * (4 - 2x).
b. To find the derivative of f(x) = (1 + 2x + x^3)^(1/4), we can rewrite it as g(x)^(1/4), where g(x) = 1 + 2x + x^3.
Step 1: Find the derivative of g(x)
g'(x) = d/dx (1 + 2x + x^3)
= 2 + 3x^2
Step 2: Apply the chain rule to find the derivative of f(x)
f'(x) = d/dx [(1 + 2x + x^3)^(1/4)]
= (1/4)(1 + 2x + x^3)^(-3/4) * d/dx (1 + 2x + x^3)
= (1/4)(1 + 2x + x^3)^(-3/4) * (2 + 3x^2)
Therefore, the derivative of f(x) = (1 + 2x + x^3)^(1/4) is f'(x) = (1/4)(1 + 2x + x^3)^(-3/4) * (2 + 3x^2).