Using the rules of differentiation, find the derivative of each of the following functions a) y=2x^3 b) f(x) = 2/x^2 c) f(x) = 4sqrt...

Using the rule of differentiation, for the first function, where

a) y=2x^3 - 7x +5, we could write dy=(2x^3 - 7x +5)dx (you could read this in this way, the derivative of the function from the left side of the equal is made  having y as unknown and the derivative of the function from the right side of the equal, 2x^3 - 7x +5, is made having x as unknown. We could write as (y)'=(2x^3 - 7x +5)')

dy=(2x^3 - 7x +5)dx=(2x^3)dx+( - 7x)dx +(5)dx

(2x^3) is a power function and it's derivative is

(2x^3)'=2*3*x^(3-1)=6*x^2

(- 7x) is a linear function, where it's derivative is

( - 7x)'=-7*1*x^( 1-1)=-7

+5 is a constant functio, where it's derivative is 0

So,  dy=(6*x^2-7)dx

b) f(x) = 2/x^2

df(x)=d(2/x^2)

 2/x^2 could be seen as a ratio and the derivative of a ratio is: the numerator derivative multiplied with the denominator minus the denominator derivative multiplied with the numerator, all these divided to the denominator square raised.

So  2/x^2=[ (2)'*(x^2)-(2)*(x^2)']/(x^2)^2

 2/x^2=-4x/x^4

Simplifying the unknown x both the numerator and denominator, we'll obtain:

 d( 2/x^2)=(-4/x^3)dx

c)f(x) = 4sqrt x

df=d(4sqrt x)

sqrt x= 1/2sqrt x, sqrt x could be seen as a power function, too; sqrt x= (x)^1/2, so, [(x)^1/2]'=1/2*(x)^(1/2-1)=

=1/2*(x)^(-1/2)=(1/2)/(x)^1/2=(1/2)/sqrt x=1/2sqrt x

d(4sqrt x)=(4sqrt x)'=4*(sqrtx)'=4*(1/2sqrt x)=2/sqrt x.

it's improper to have a square root at denominator, so, we'll amplify with the same value of the square root, in order to have a denominator without square root.

d(4sqrt x)=(2*sqrt x/x)

d)g(L) = 2pi sqrt L/9.8

d(g(L))= d(2pi sqrt L/9.8)

If you are looking at the function above, the only difference is that the unknown is L and not x, the unknown we're used to. You can re-write the function g(L) in a more  intelligible way, more accurate, so g(L)= (2pi/sqrt 9.8)*sqrtL.

All you have to do is to consider sqrtL as  sqrt x=1/2sqrt x, so sqrt L=1/2sqrtL

d(g(L))=[(2pi/sqrt 9.8)*sqrtL]'=(2pi/sqrt 9.8)*1/2sqrtL=

d(g(L))=  (pi *sqrtL/L*sqrt 9.8)dx