Question

Using the definition of the derivative to differentiate.?

Use the definition of the derivative to differentiate f.

1) f(x) = x^4

2) f(x) = 1/(x^3)

Show step by step please. Thank you. =)

Answers

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#1

Using the difinition of derivative.

............f(x+h)-f(x)

f'(x)=lim--------------------

.......h-->0...h

f(x)=x^4

1). f(x+h)=(x+h)^4

=h^4+4 h^3 x+6 h^2 x^2+4 h x^3+x^4

..........h^4+4 h^3 x+6 h^2x^2+4hx^3+x^4-x^4

=lim------------------------------------…

...h->0....................h

..........h^4+4h^3x+6h^2x^2+4hx^3

=lim----------------------------------…

.h->0....................h

=lim h^3+4h^2x+6hx^2+4x^3

..h->0

=lim (0)^3+4(0)^2x+6(0)x^2+4x^3

.h->0

=4x^3 answer//

2) f(x) = 1/(x^3)

..............1

f(x+h)=-------------

............(x+h)^3

................1

=-------------------------------------…

...h^3+3h^2x+3hx^2+x^3

................1.....................…

f'(x)=--------------------------------… - ---------------

.h->0.h^3+3h^2x+3hx^2+x^3..........x^3

--------------------------------------…

.........................h

.=lim ..........x^3- (h^3+3h^2x+3hx^2+x^3)

..h->0 ----------------------------------------…

.............x^3(h^3+3h^2x+3hx^2+x^3)

...........---------------------------…

.......................h

........ x^3-h^3-3h^2x-3hx^2-x^3

=lim----------------------------------…

..h->0..x^3(h^3+3h^2x+3hx^2+x^3)

...........---------------------------…

..........................h

.........-h^3 - 3h^2x - 3hx^2

=lim ---------------------------------

.h->0.x^3(h^3 + 3h^2x + 3hx^2 + x^3)..................x^3(h^3+3h^2x+3hx^…

....----------------------------------… x---------------------------------------…

.................h....................… + 3h^2x + 3hx^2 + x^3)

...

=lim .........-h^3-3h^2x-3hx^2

.h->0---------------------------------…

........hx^3(h^3+3h^2x+3hx^2+x^3)

........-h(h^2+3hx+3x^2)

=lim----------------------------------…

.h->0 hx^3(h^3+3h^2x+3hx^2+x^3)

...........-(h^2+3hx+3x^2)

=lim----------------------------------…

h->0..x^3(h^3+3h^2x+3hx^2+x^3)

.........-(0^2+3(0)x+3x^2)

=lim----------------------------------…

h->0 x^3((0)^3+3(0)^2x+3(0)x^2+x^3)

.....-3x^2

=-------------------

....x^6

.....-3

=-------- answer//

....x^4

#2

this is a very simple question

f(x) = x^4

f`(x) = 4 x^3

f``(x) = 12 x^2

follow the formula d/dx (X) = n * x ^ n-1

the second one

take the denominator on to the top as

f(x) = 1/(x^3)

= x ^ -3

then applying the formula

= -3 x ^ -4 [ as in the power we get -3 -1 ]

f``(x) = 12 x ^ -5

= 12 / x^5

thats it !!

#3

1) f(x) = 4x^3 .................. X^n= nX^n-1

2) f(X)= X^-3

= -3X^-4

= -3/( X^4)

#4