Question

Find the derivative of each function defined as follows.
y=12^3-8x^2+7x+5

Answer 1

\[y=12x^3-8x^2+7x+5\]

Answer 2

hint:

\[\Large y = x^n\]

\[\Large y^{\prime} = nx^{n-1}\]

Answer 3

So for example

\[\Large y = x^7\]

\[\Large y^{\prime} = 7x^6\]

Answer 4

i dont understand

Answer 5

Another example

\[\Large y = 3x^8\]

\[\Large y^{\prime} = 8*3x^{8-1}\]

\[\Large y^{\prime} = 24x^{7}\]

Answer 6

can you help me step by step with this question

Answer 7

This is the power rule. It says "to derive x^n, you pull down the exponent n and multiply it with the coefficient and then subtract the exponent"

Check out this page

http://www.mathscoop.com/calculus/derivatives/the-power-rule.php

Answer 8

So the first one will be 36x^2

Answer 9

that is correct

what about the next one?

Answer 10

-16x

Answer 11

good

Answer 12

how about the next one

Answer 13

7x

Answer 14

or 7

Answer 15

7x = 7x^1

so the derivative of 7x^1 is 1*7x^0 = 7*1 = 7

Answer 16

So put all together, the derivative of

\[\Large y=12x^3-8x^2+7x+5\]

is

\[\Large y=36x^2-16x+7\]

Answer 17

what happens to the 5

Answer 18

the derivative of any constant is 0

Answer 19

so it effectively goes away

Answer 20

okay now whats next

Answer 21

that's it, the final answer is posted above

Answer 22

oh okay i get it. Thank you very much. Ill contact you if i get stuck with a problem.

Answer 23

how do i turn this into a derivative x/12

Answer 24

@jim_thompson5910

Answer 25

\[\Large y = \frac{x}{12}\]


\[\Large \frac{d}{dx}[y] = \frac{d}{dx}\left[\frac{x}{12}\right]\]


\[\Large \frac{d}{dx}[y] = \frac{d}{dx}\left[\frac{1}{12}*x\right]\]


\[\Large \frac{d}{dx}[y] = \frac{1}{12}\frac{d}{dx}\left[x\right]\]


\[\Large \frac{d}{dx}[y] = \frac{1}{12}\frac{d}{dx}\left[x^{1}\right]\]


\[\Large \frac{d}{dx}[y] = \frac{1}{12}*1*x^{1-1}\]


\[\Large \frac{d}{dx}[y] = \frac{1}{12}*1*x^{0}\]


\[\Large \frac{d}{dx}[y] = \frac{1}{12}*1*1\]


\[\Large \frac{d}{dx}[y] = \frac{1}{12}\]


\[\Large y^{\prime} = \frac{1}{12}\]

Answer 26

\[y=8\sqrt{x}+6x ^{3/4}\]

Answer 27

@jim_thompson5910

Answer 28

I need help with this one

Answer 29

rewrite \[\Large \sqrt{x}\] as \[\Large x^{1/2}\]

Answer 30

then use the power rule to derive

Answer 31

\[\Large y=8\sqrt{x}+6x ^{3/4}\]


\[\Large \frac{d}{dx}[y]=\frac{d}{dx}[8\sqrt{x}+6x ^{3/4}]\]


\[\Large y^{\prime}=\frac{d}{dx}[8x^{1/2}]+\frac{d}{dx}[6x^{3/4}]\]


\[\Large y^{\prime}=8\frac{d}{dx}[x^{1/2}]+6\frac{d}{dx}[x^{3/4}]\]


\[\Large y^{\prime}=8*\frac{1}{2}*x^{1/2-1}+6*\frac{3}{4}*x^{3/4-1}\]


\[\Large y^{\prime}=4*x^{-1/2}+\frac{9}{2}*x^{-1/4}\]


\[\Large y^{\prime}=\frac{4}{x^{1/2}}+\frac{9}{2x^{1/4}}\]


\[\Large y^{\prime}=\frac{4}{\sqrt{x}}+\frac{9}{2\sqrt[4]{x}}\]


You can optionally rationalize the denominator, but that's not necessary in my opinion.