Question

compute the first and second derivative of the function f(x)=(5x^2 +x)^4

Answer 1

f(x)=g(h(x)) => f'(x)=h'(x)g'(h(x))
=> f''(x)=h''(x)g'(h(x))+h'(x)h'(x)g''(h(x))

Answer 2

okay so is f'(x)=4(5x^2+x)^3(10x)+1

Answer 3

almost the paranthesis should be around 10x+1

Answer 4

not just the 10x

Answer 5

okay so f'(x)=4(5x^2+x)^3 (10X+1)

Answer 6

yes except that X should be a little x but you knew that

Answer 7

yes, thank you. for f"(x) I got 12(5x^2+x)^2 (10x+1) (10), is this correct

Answer 8

you will need product rule to find the derivative of f'

Answer 9

we have h' times g'(h(x))
applying product rule:
h'' times g'(h(x)) + h' times h' g''(h(x))

/\
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here I used chain rule
to find the derivative of
g'(h(x))
you remember
derivative of inside times
derivative of outside

Answer 10

okay

Answer 11

\[f'=[4(5x^2+x)^3](10x+1)\]
\[f''=[4(5x^2+x)^3]'(10x+1)+[4(5x^2+x)^3](10x+1)'\]

Answer 12

\[[4(5x^2+x)^3]'=4 \cdot 3 (5x^2+x)^{3-1} (5x^2+x)'=12(5x^2+x)^2(10x+1)\]

Answer 13

\[(10x+1)'=(10x)'+(1)'=10+0\]

Answer 14

okay, thank you very much

Answer 15

no problem you understand to plug those last two lines of mine into the expression i wrote for f'' right?

Answer 16

yes, but would for [4(5x^2+x)^3]' would it be 12(5x^2+x)^2 (10x+1) (10) or not

Answer 17

\[[(5x^2+x)^n]'=n(5x^2+x)^{n-1}(5x^2+x)' \]
so no

Answer 18

\[[h(g(x))]'=h'(g(x)) \cdot g'(x)\]

Answer 19

the outside function is h(x)=x^3 and inside function is g(x)=5x^2+x
h'(x)=3x^2 g'(x)=10x+1
h'(g(x))=3(5x^2+x)^2
so \[[h(g(x))]'=[(5x^2+x)^3]'=3(5x^2+x)^2(10x+1)\]

Answer 20

okay, thanks

Answer 21

np have a good day

Answer 22

thanks, you too