Question

f(x) = 3 x^{6} \sqrt{x} + \frac{-3} {x^{2} \sqrt{x} }.
how do you find the derivative

Answer 1

\[f(x) = 3 x^{6} \sqrt{x} + \frac{-3} {x^{2} \sqrt{x} }\]?

Answer 2

Just apply sum and power rule.

Answer 3

yes but i dont know how to do this

Answer 4

okay so use the product rule for the first half and then use the quotient rule for the second half

Answer 5

\(x^n\sqrt x = x^{n+1/2}\)

Answer 6

Nah, we only need power rule

Answer 7

So... \[\begin{align}
3 x^{6} \sqrt{x} + \frac{-3} {x^{2} \sqrt{x} } &= 3 x^{13/2} - \frac{3} {x^{5/2}}\\~\\&= 3 x^{13/2} - 3x^{-5/2}
\end{align}\]

Now apply power rule: \(\dfrac{d}{dx}x^n = nx^{n-1}\)

Answer 8

what is n

Answer 9

Any number

Answer 10

While taking derivative of x, you just multiply it by its exponent, then subtract its exponent by 1.

Answer 11

yes but if i put that as my answer it comes out wrong

Answer 12

You don't put \(d/dx~x^n = nx^{n-1}\) as answer... It's just power rule you need to apply.

Answer 13

so how can i find the answer

Answer 14

Let's do a few easy questions. What is derivative of \(x^2\)?

Answer 15

2x

Answer 16

Yes, what about \(x^3\)?

Answer 17

i know how to apply the power rule

Answer 18

ok, what are you struggle with?

Answer 19

3x^2

Answer 20

to find the answer

Answer 21

Well, just take derivative, then simplify

Answer 22

Remember sum rule; basically you just derivative each term individually.

Answer 23

so its 13/3 times 3

Answer 24

13/2 times 3. Times x with what exponent?

Answer 25

i dont follow

Answer 26

\(3*\dfrac{13}{2}x^?\)

Answer 27

dont you subtract 13/2-1

Answer 28

Yeah, so \(3*\dfrac{13}{2}x^{11/2}\)

Now do next term. can you find derivative of that term?

Answer 29

7.5/x^7/2?

Answer 30

Yeah, so you have \(\dfrac{d}{dx}f(x) = \boxed{\dfrac{39}{2}x^{11/2} + \dfrac{15}{2x^{7/2}}}\)

Answer 31

how did you get 15

Answer 32

\(7.5 = \dfrac{15}{2}\)

Answer 33

Thank you so much I appreciate it

Answer 34

wait

Answer 35

how did you get x^5/2 in the begininng

Answer 36

and 3x^13/2

Answer 37

\(x^2\sqrt x = x^{2+\frac{1}{2}}=x^{\frac{4}{2}+\frac{1}{2}} = x^{5/2}\)

Same for \(x^{13/2}\)

Answer 38

Thank you I get it