Question

If d/dx(f(3x^2))=9x^3, calculate f′(x)

Answer 1

First I'm going to just focus on the left side

Use the chain rule to get

d/dx[f(3x^2)] = 9x^3
f ' (3x^2)*d/dx[3x^2] = 9x^3
f ' (3x^2)*6x = 9x^3

then divide both sides by 6x to get

f ' (3x^2) = 1.5x^2

with me so far?

Answer 2

yes

Answer 3

Let

g(x) = f ' (x)


we know that


g(3x^2) = 1.5x^2


what we want to find is g(x)


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Let's assume g(x) is a linear function. So g(x) = ax+b



g(x) = ax+b
g(3x^2) = a(3x^2)+b
g(3x^2) = 3ax^2+b


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we know that g(3x^2) = 1.5x^2 and we found based on the linear assumption that g(3x^2) = 3ax^2+b, so let's equate the two


1.5x^2 = 3ax^2+b
1.5x^2+0 = 3ax^2+b

we see that

1.5x^2 = 3ax^2
0 = b

so we know b = 0

let's find the value of 'a'

1.5x^2 = 3ax^2
1.5 = 3a
3a = 1.5
a = 1.5/3
a = 0.5


So g(x) = 0.5x+0 = 0.5x


This means f ' (x) = 0.5x

Answer 4

thanks so much

Answer 5

no problem