Definition – Differentiation is the rate of change of the function value, f(x) to the change in the value of ‘x’.Finding a derivative of any function is the same thing as finding the slope at a particular point. Hence derivative is also called as the slope of the tangent drawn at any point on any curve!
Example: Given f(x) = 3x, find its derivative using the above formula.
So, dy/dx = [f(x+∆x) – f(x)] / ∆x
-> dy/dx= [3(x+∆x) – 3x] / ∆x
-> dy/dx= (3x +3∆x-3x) / ∆x
-> dy/dx= 3∆x / ∆x
-> dy/dx=3 meaning derivative of function, f(x)=3x is ‘3’
Power rule of Derivatives:
Example: What is the derivative of x4?
Here n=4, so dy/dx= 4x4-1
Example: What is the derivative of 1/x3?
Here 1/x3 can also be written as x-3, so n=-3
dy/dx=-3x-3-1 = -3x-4
Example: What is the derivative of 3x2+x5?
Using the Sum rule, (f+g)’=f’ + g’
-> d (3x2+x5)/dx=3*2x2-1 +5*x5-1
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