Introduction
Let ‘f’ be a given function, then the derivative of ‘f’ is denoted by f’(x) and is defined as,
The process of finding derivative is called differentiation.
The derivative of a function can be denoted is different ways, they are
y’, y1, dy / dx etc.
The derivative of a function at ‘c’ is denoted as f’(c) and is defined as
The process of finding the derivative using definition is called the first principle of differentiation
Example: Using 1st principle of differentiation, find the derivative of (x+2)2
Let f(x) = (x + 2)2, f(x + h) = (x + h + 2)2
f’(x)
= 0 + 2x + 4
= 2(x + 2)
List of derivatives of certain standard functions
Product Rule of Differentiation
If ‘u’ and ‘v’ are functions of ‘x’ then
Derivative of product of two functions is “(first function) x (derivative of second) + (second function) x (derivative of first)”
If u, v and w are functions of ‘x’ then
Quotient Rule of Differentiation
If ‘u’ and ‘v’ are functions of ‘x’ and v≠0, then quotient rule is
Important Notes:
(i) (u ± v)’ = u’ ± v’
(ii) If a function ‘f’ is differentiable at a point ‘c’ then it is continuous at that point.
(iii) Every differentiable function is continuous.
Chain Rule of Differentiation
Chain Rule is applicable only for the composition of functions. Let ‘y’ be a composition of two functions ‘f’ and ‘g’.
y = f o g = f [g(x)]
Take y= f(u) where u = g(x) so that we can find
dy/du and du/dx [Since ‘y’ is a function of ‘u’ we get dy/du and ‘u’ is a function of ‘x’ we get du/dx]
Hence
If ‘y’ is the composition of three functions ‘f’, ‘g’ and ‘u’ then
y= f o g o u
= f {g[ u(x) ]}
Take v=u(x), t=g(v) and y=f(t)
Find dv/dx, dt/dv and dy/dt
Example: Find the derivative of Cos (3x + 5)
Solution: Let y = Cos (3x + 5)
Take y = Cos (u) where u = 3x + 5
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