Derivative Problem

Question

Find the derivative of the function 

Answer 1

To find the derivative, we will apply the chain rule. It states: if f(x) = g(h(x)) then f'(x) = g'(h(x))*h'(x) In other words, the derivative of f(x) is equal to the original inner function (h) plugged into the derivative of the outer function (g') multiplied by the derivative of the inner function (h'). You can consider f(x) = (2x^2-x+4)^3 to be a composite function. if f(x) = g(h(x)) then g(x) = x^3 [outer] and h(x) = 2x^2-x+4 [inner] Note: The "x" in g(x) is really h(x) as it is written out in f(x)'s full form (the original function). We'll need the derivatives of both g(x) and h(x). They can be easily differentiated using the power rule. g'(x) = 3x^2 h'(x) = 4x-1 Now, we apply the chain rule: f'(x) = 3(2x^2-x+4)^2 * (4x-1) Simplifying, we get f'(x) = 48x^5 - 60x^4 + 216x^3 - 147x^2 + 216x - 48 Having finished the problem, I think it might be faster to expand (2x^2-x+4)^3 and use the power rule on the expansion... Either way, this problem requires an excessive amount of polynomial multiplication. Tell your teacher to give more reasonable examples!

Answer 2

{ Let's try that again, with proper formatting. }

To find the derivative, we will apply the chain rule. It states:

if f(x) = g(h(x))

then f'(x) = g'(h(x))*h'(x)

In other words, the derivative of f(x) is equal to the original inner function (h) plugged into the derivative of the outer function (g') multiplied by the derivative of the inner function (h').

You can consider f(x) = (2x²-x+4)³ to be a composite function.

if f(x) = g(h(x))

then g(x) = x³ [outer]

and h(x) = 2x²-x+4 [inner]

Note: The "x" in g(x) is really h(x) as it is written out in f(x)'s full form (the original function).

We'll need the derivatives of both g(x) and h(x). They can be easily differentiated using the power rule.

g'(x) = 3x²

h'(x) = 4x-1

Now, we apply the chain rule:

f'(x) = 3(2x²-x+4)² * (4x-1)

Simplifying, we get

f'(x) = 48x⁵ - 60x⁴+ 216x³ - 147x² + 216x - 48

Having finished the problem, I think it might be faster to expand (2x²-x+4)³ and use the power rule on the expansion...

Either way, this problem requires an excessive amount of polynomial multiplication. Tell your teacher to give more reasonable examples!

Answer 3

3(2x^2-x+4)^2  * 4x-1

then simplify it

(6x^2-3x+12)^2  * 4x-1