Derivative

Differentiate

f(x)=(5x-4)^2

There are two ways to do this:
(1) let u = 5x-4, and use the "chain rule", df/dx = df/du * du/dx
OR
(2) expand the polynomial to
f(x) = 25 x^2 -40x + 16
and differentialte it term-by-term.
You will get the same answer either way.

Let's use the first way, because it is something you may need for other problems
u = 5x-4
du/dx = 5
f(u) = u^2
df/dx = df/du * du/dx
= 2 u * 5 = 10(5x -4) = 50x - 40

To differentiate the function f(x) = (5x-4)^2, we can use the chain rule. The chain rule states that if we have a composition of functions, we can differentiate them by taking the derivative of the outer function with respect to the inner function multiplied by the derivative of the inner function.

1. Let u = 5x-4. This is our inner function.
2. Find du/dx, which is the derivative of u with respect to x. Since u = 5x-4, du/dx = 5. This is the derivative of the inner function.
3. Rewrite our function f(x) = (5x-4)^2 as f(u) = u^2. This is the outer function.
4. Differentiate f(u) = u^2 with respect to u. The derivative of u^2 is 2u.
5. Multiply df/du by du/dx. df/du = 2u and du/dx = 5x-4.
6. Simplify the expression. df/dx = df/du * du/dx = 2u * 5 = 10(5x - 4) = 50x - 40.

Therefore, the derivative of f(x) = (5x-4)^2 is df/dx = 50x - 40.