Fundamental Rules for Differentiation
( v ) if d / d ( x ) f ( x ) = φ ( x ), then d / d ( x ) f ( ax + b ) = a φ ( ax + b ) ( vi ) Differentiation of a constant function is zero i . e ., d / d ( x ) ( c ) = 0 . Geometrically Meaning of Derivative at a Point
Geometrically derivative of a function at a point x = c is the slope of the tangent to the curve y = f ( x ) at the point { c , f ( c )}.
Slope of tangent at P = lim x → c f ( x ) – f ( c ) / x – c = { df ( x ) / d ( x )} x = c or f ’ ( c ). Different Types of Differentiable Function 1 . Differentiation of Composite Function ( Chain Rule )
If f and g are differentiable functions in their domain , then fog is also differentiable and
( fog )’ ( x ) = f ’ { g ( x )} g ’ ( x ) More easily , if y = f ( u ) and u = g ( x ), then dy / dx = dy / du * du / dx .