LIMITS AND
DERIVATIVES
SOME STANDARD DERIVATIVES
ALEGBRA OF DERIVATIVES OF FUNCTION
DERIVATIVE OF A FUCTION AT A
LIMIT
ALGEBRA OF LIMITS
SOME STANDARD LIMITS
DERIVATIVE OF POLYNOMIAL FUCTION
FOR FUNCTION U AND V THE FOLLOWING HOLDS:
• (u ± v)= u ± v`
• (u v)= uv + v`u
• (u/v)= uv – uv/v provided all are defined and v ≠0
• u= du/dx and v = dv/dx
Let f(x)= anxn+an-1xn-1+....+a+a1x+a0 be a polynomial function where ais are all the real numbers and an ≠ 0. Then the derivatives functions is given by df(x)/dx =nanx-1 + (n-1)an-1xn-2 + …. +2a2x + a1
The derivative of e function of at a is defined by
f`(a) = lim (h=0) f(a+h) – f(a)/h
eg=> find derivative of f(x)=1/x
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lim (h=0) (1/x+h) – (1/x)/h
lim (x=0) 1/h [-h/x(x+h)] = -1/x^2
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sol=> f`(x) = lim (h=0) f(x+h) – f(x/h)
Lim(x=a) x^n -a^n/x – a =na^n-1
Lim(x=0) sinx/x =1
Lim(x=0) 1-cosx/x =0
For functions f and g the following holds
Lim(x=a) [f(x) ± g(x)] = lim(x=a) f(x) ± lim(x=a) g(x)
Lim(x=a) [f(x).g(x)] = lim(x=a) f(x).lim(x=a) g(x)
Lim(x=a) [f(x)/g(x)] = lim(x=a) f(x)/lim(x=a) g(x) , ∴ lim(x=a) g(x) ≠ 0
We say lim(x=a) f(x) is the expected value of f at x=a given that the value of f near x to the left of a . this value is called the left hand limit of f ata.
We say lim(x=a) f(x) is the expected value of f at x=a given that the value of f near x to the right of a . This value is called the right hand limit of f at a.
If the right and left hand limits coincide , we call that common value as the limit of f(x) at x=a and denote it by the lim(x=a) f(x).
Eg=> find the limit of the function f(x) = (x-1)^2 at x=1
Sol=> left hand limit (LHL) (at x=1) = lim(x=1) (x-1)^2=0
Right hand limit (RHL) (at x=1) = lim(x=1) (x-1)^2 = 0
∴ LHL =RHL
d/dx(sinx) = cosx
d/dx (cosx) = -sinx
d/dx(x^n)=nx^n-1