Question

derivative of 3x - 5 cos (pi x)^2

Answer 1

you know the derivative of 3x right? no problem there?

Answer 2

yes , that is 3 but how about the second part how will i am going to perform that

Answer 3

\[-5[\cos (\pi x)]^2 \implies -5\cos^2 (\pi x)\]
first of all..you can pull out -5 since it's a constant
\[-5( \frac{d}{dx} \cos^2 (\pi x))\]
do you know the derivative of cos^2 pi x? or no?

Answer 4

Differentiate the sum term by term and factor out constants

Answer 5

\[\large{3(\frac{d}{dx}(x))-5\frac{d}{dx}(cos^2(\pi x))}\]

Answer 6

Well sorry for inturruption ..@lgbasallote you may continue .. sorry

Answer 7

why did you move the exponent to cos, is that the rule if in the given cos^2 we will move then.. derivative of cos^2 is -2sin?

Answer 8

well it's not a rule...but it's an option..\[(\cos x)^2 = \cos^2 x\]

Answer 9

i just moved it so it's easier...

Answer 10

anyway...derivative of cos^2 is not -2 sin...you have to use power rule..

Answer 11

@lgbasallote it will be better I think if you will not move the exponent..

Answer 12

hmm ok..

Answer 13

I mean to say that it will become easy to use power rule there just like in case of \(x^2\)..

Answer 14

\[[\cos (\pi x)]^2\]
first do power rule (which is \(\frac{d}{dx} (x^n) = nx^{n-1}\))

so \[\implies 2[\cos (\pi x)]^{2-1}\]
do you get this part so far?

Answer 15

yes

Answer 16

now...that's not yet finished...according to chain rule...you now have to take the derivative of \[\cos (\pi x)\]
now..what is the derivative of this?

Answer 17

-sin(πx), wait is π a constant?

Answer 18

yes pi is a constant

Answer 19

so you're right...now according to chain rule...you now haveto take the derivative of \[(\pi x)\]

Answer 20

π?

Answer 21

correct. so now you multiply everything together
\[-5[2\cos (\pi x) \times -\sin (\pi x) \times \pi]\]
what will this give you?

Answer 22

10πcos(πx)sin(πx)

Answer 23

right!

Answer 24

you can also do it like this \[\implies 5 \pi [2\sin (\pi x) \cos (\pi x)]\]
\[\implies 5\pi \sin (2\pi x)\]
because of the trig identity \(2\sin a \cos a = \sin (2a)\)

Answer 25

is that the final answer?
including the 3

Answer 26

yup

Answer 27

oh wait , how can we go to 10π^2xsin(πx)^2

Answer 28

what do you mean?

Answer 29

3+10π^2xsin(πx)^2
that is the answer po. ?

Answer 30

can you draw it? \[10 \pi^2 x \sin (\pi x)^2?\]

Answer 31

no i cant , that is the answer ksi given on the book?

Answer 32

uhh wait... the qustion is \[\Large 3x - 5\cos (\pi x)^2\]
or \[\Large 3x - [5\cos (\pi x)]^2\]

Answer 33

the first one po

Answer 34

hmm i thought so.... i just solved the second one =_=

Answer 35

okay..here's how to do the first one...

Answer 36

first..pull out -5 then take the derivative of \(\cos (\pi x)^2\)

Answer 37

\[\frac{d}{dx} [\cos (\pi x)^2] \implies -\sin (\pi x)^2 \times 2(\pi x) \times \pi\]
do you follow how i got that? or do i need to explain it more?

Answer 38

I understand it

Answer 39

great.. so \[-5[-\sin(\pi x)^2 \times 2(\pi x) \times pi] \implies - 5[ -\sin (\pi x)^2 \times 2\pi^2 x]\implies 10\pi^2 x \sin (\pi x)^2\]
got it?

Answer 40

sorry for screwing it up a while ago =))

Answer 41

no need for you to say sorry po, I should be the one who apologize, anyways thank you po for helping me. :)

Answer 42

welcome ^_^