Question

Calculate the derivative of the given function f(x)=8sin^3(√x)
a) f′(x)=24sin^2(√x)cos(√x)
b) f′(x)=48sin^2(√x)cos(√x)/x√x
c) f′(x)=24cos(√x)
d) f′(x)=12sin^2(√x)cos(√x)/√x
e) f′(x)=12cos(√x)/√x

Answer 1

yes, one second

Answer 2

Sorry, I made a typo. \[f(x) = 8\sin^3(x^{1/2})\]

Answer 3

actually could you help me with this really quickly? its more pressing:
Evaluate (g∘f)′(6), given that f(4)=4, f(5)=6, f(6)=6, g(4)=5, g(5)=5, g(6)=4, f′(4)=6, f′(5)=5, f′(6)=4, g′(4)=5, g′(5)=5, g′(6)=4
a) 16
b) 19
c) 18
d) 17
e) 15

Answer 4

sorry, thanks for all your help, im just on a time crunch.

Answer 5

You're not taking a test are you?

Answer 6

no, I just have to be somewhere soon

Answer 7

Oh, alright.

Answer 8

And this homework is due tomorrow

Answer 9

So i assume : \[(g~\circ ~ f)'(x) \qquad \implies \qquad \frac{d}{dx}(g(f(x)) = g'(f(x)) \cdot f'(x)\]

Answer 10

where \(x=6\)

Answer 11

Ack, I've got to head off for a little while, I will be back very soon.

Answer 12

@DanJS @jim_thompson5910 mind taking over? Thanks!

Answer 13

i am late, are you still working on these things?

Answer 14

yes, that one would be just the chain rule, and they list the values you may need for the functions and their derivatives
\[\qquad \frac{d}{dx}(g(f(6)) = g'(f(6)) \cdot f'(6) \]

Answer 15

f(6) = 6
f '(6) = 4
g'(6)=4
so
\[\qquad \frac{d}{dx}(g(f(x)) = g'(f(x)) \cdot f'(x) = g'(6)*f'(6) = 4*4\]

Answer 16

To find f'(x) you use the chain rule...

\[\frac{ d }{ dx }f(x) = \frac{ d }{ du }*\frac{ du }{ dx } f(x)\]

Answer 17

If you have a function inside a function , like f(g(x))

f(x) = sin(x^3)

I usually think something like;

" f ' (x) is equal to derivative of sin(inside function left alone) times the derivative of the inside function"

f ' (x) = cos(x^3) * 2x^2

thats it, goodluck

Answer 18

similar with the given function

f(x) = 8*sin(x^(1/2))^3

start with outside and leave inside function same,
except here you have to use the rule two times, u^3, sin(u) and x^(1/2)
start from outside power function

\[f ' (x) = 3*8*[\sin(x^{1/2})]^2~*~\cos(x^{1/2})*\frac{ 1 }{ 2 }*x^{-1/2}\]

Answer 19

simplifies to D i think ... check it out,

Answer 20

keep leaving the inner functions alone, and work outside to inside, power rule - derivative of sin() - derivative of root x

Answer 21

\[[f(g(h(x)))] ' = f'(g(h(x))) *g'(h(x)) * h'(x)\]

Answer 22

you can have as many nested functions as you want, same idea