Question

find derivative of
h(x)= (3x^2-5x+2)(7x+4)^2
use product rule,
please show process

Answer 1

product rule:

f'(x) = u'v + v'u

In this case let u = 3x^2 -5x + 2 then u' = 6x-5
v= (7x+4)^2 so v'= 2(7x+4)(7) = (14x+6)(7) = 98x+ 42.

Plug them in the formula given, you will find the first derivative. All the best.

Answer 2

Whenever I see people who don't believe the power rule works, I suggest they do this simple little thought experiment:

f(x)=x^5=(x^2)(x^3)
f'(x)=5x^4=(2x)(x^3)+(3x^2)(x^2)
=(2x^4)+(3x^4)
=5x^4

And then just do that with anything you don't really understand just so that you get back an answer of something you already know is true so that you know you're doing it the right way rather than just believing your teacher for no reason.

Answer 3

Product and chain rule for this one.

Answer 4

\[h(x)= (3x^2-5x+2)(7x+4)^2\]
\[h'(x) = (6x-5)(7x+4)^2 + (3x^2-5x+2)2(7x+4) * 7\]
\[h'(x) = (6x-5)(7x+4)^2 + 14(3x^2-5x+2)(7x+4)\]

Answer 5

The product and chain rule combined is :
\[f'(x)g(h(x)) + f(x)g'(h(x)) * h'(x)\]

Answer 6

\[\large \text{Let f(x):}\]
\[\large (3x^2-5x+2)\]
\[\large \text{Let g(h(x)):}\]
\[(\large 7x+4)^2\]
\[\large \text{Let h(x):}\]
\[\large 7x+4\]
Then you would use the combined product + chain rule I showed above..