 Hi and welcome to the session. Today we will learn about differentiability. Suppose f is a real function, b is a point in its domain then derivative the function f at point c is defined by limit h tends to 0 f of c plus h minus f of c upon h provided this limit exists. If limit does not exist then that means f is not differentiable at point c. In other words we can say that the function f is differentiable at a point c in its domain if both left hand limit and right hand limit are finite and are equal to each other. Now derivative of the function f at point c is denoted by f dash of c or d by dx of f of x at point c. Now derivative of f is defined by the function f dash of x is equal to limit h tends to 0 f of x plus h minus f of x upon h. Wherever the limit exists also the function f is differentiable in the closed interval a comma b or the open interval a comma b the function f is differentiable at every point of the closed interval a comma b or the open interval a comma b respectively. Now there are few rules of differentiation which are as follows for the functions u and v. First rule is differentiation of u plus v is equal to differentiation of u plus differentiation of v. Second is differentiation of u minus v is equal to differentiation of u minus differentiation of v. Third rule is differentiation of u into v is equal to differentiation of u into v plus u into differentiation of v. This is known as product rule and the fourth one is differentiation of u upon v is equal to differentiation of u into v minus u into differentiation of v upon v square wherever v is not equal to 0. This is known as quotient rule. Let's see the differentiation of some standard functions differentiation of x to the power n is equal to n into x to the power n minus 1 differentiation of sin x is equal to cos x differentiation of cos x is equal to minus sin x and differentiation of tan x is equal to secant square x. Now we have important result which says that if a function f is differentiable at a point c it is also continuous that every differentiable function is continuous. Now our next topic is derivatives of composite functions and to find out the derivatives of composite functions we use chain rule. Let f is a real valued function which is a composite of two functions u and v that is f is equal to v or u and suppose t is equal to u of x and if both dt by dx dv by dt exists then of the function f with respect to x that is df by dx is equal to first we differentiate the function v with respect to t that is dv by dt into now we will differentiate t with respect to x that is dt by dx let's take one example here we are given the function f of x equal to sin 5x plus 3 and we need to find f dash of x that is derivative of f of x so here we will take u of x as 5x plus 3 and v of t equal to sin t and we will put t equal to u of x so the function f of x is equal to v of u of x by dx is equal to dv by dt into dt by dx by chain rule by dt is equal to cos t dt by dx is equal to df by dx will be equal to cos t into 5 and this will be equal to 5 into s t by u of x that is 5x plus 3 differentiation of the function f of x so with this we finish this session hope you must have understood all the concepts goodbye