 Hello and welcome to the session. In this session we will discuss derivatives. In many cases we need to know how a particular parameter is changing with respect to some other parameter. For this purpose we use derivatives. Derivative of a function fx at a is denoted by f dash a and this is equal to limit h tends to 0 f of a plus h minus f of a upon h, where f is a real valued function and a is the point in its domain of definition. We observe that f dash a quantifies the change in fx at a with respect to x. Consider the function fx equal to 2x. We need to find the derivative of fx at x equal to 2 that is we need to find f dash 2. Here we have a is equal to 2. Now we have f dash 2 is equal to limit f tends to 0 f of a plus h minus f of a upon h. This becomes equal to limit h tends to 0 2 into 2 plus h minus 2 into 2 upon h that is equal to limit h tends to 0 4 plus 2h minus 4 upon h that is we have limit h tends to 0 2h upon h. Now h and h gets cancelled. So this is equal to limit h tends to 0 2 and this is equal to 2. So we have f dash 2 is equal to 2. Next we discuss first principle of derivative. Derivative of a function f at any point x is defined by f dash x which can also be written as dfx upon dx. This is equal to limit h tends to 0 f of x plus h minus fx upon h where again f is a real valued function. Consider fx equal to 11x. Now f dash x would be equal to limit h tends to 0 f of x plus h minus fx upon h that is this is equal to limit h tends to 0 11x plus h minus 11x upon h. This is equal to limit h tends to 0 11h upon h now h and h gets cancelled. So this is equal to limit h tends to 0 11 so we get f dash x is equal to 11. This is how we use first principle of derivative to find the derivative of a function f at any point x. Next we discuss algebra of derivative of functions. Consider u and v the two functions such that their derivatives exist and are defined in a common domain. Then we have u plus v dash is equal to u dash plus v dash that is derivative of sum of two functions is the sum of the derivatives of the functions. Then next we have u minus v dash is equal to u dash minus v dash that is derivative of difference of two functions is difference of the derivatives of the function. Next is u v dash is equal to u dash v plus u v dash derivative of product of two function is given by this product rule. Next is u by v dash is equal to u dash v minus u v dash upon v square provided all are defined this is the derivative of question of two functions and this is the question rule. Let's find the derivative of the function given by fx equal to sin x into cos x. Here we consider u equal to sin x and v equal to cos x. Then f dash x would be equal to u dash that is derivative of sin x that is cos x into v that is cos x plus u that is sin x into v dash that is derivative of cos x and that is minus sin x and this is equal to cos square x minus sin square x and this is equal to cos 2x that is f dash x is equal to cos 2x. Derivative of the function sin x into cos x is cos 2x. Then next very important result is derivative of the function fx equal to x to the power n is given by n into x to the power n minus 1 for any positive integer n. Consider fx to be equal to x to the power 5 then f dash x is equal to n that is 5 into x to the power n minus 1 which is equal to 5 into x to the power 4. This result is true for all powers of x that is n can be any real number. Next is another very important result according to which we have let fx be equal to a n x to the power n plus a n minus 1 x to the power n minus 1 plus and so on a 1 x plus a naught this be a polynomial function where a i's are real numbers and we have a n is not equal to 0 then the derivative function is given by dfx upon dx and this is equal to n into a n x to the power n minus 1 plus n minus 1 into a n minus 1 into x to the power n minus 2 plus and so on up to 2 a 2 x plus a 1. Consider the polynomial function given by fx equal to x to the power 10 minus 5 into x to the power 2 plus x. So derivative of this function fx is given by f dash x is equal to 10 into x to the power 10 minus 1 that is 9 minus 5 into 2 into x to the power 2 minus 1 that is 1 plus 1. This is equal to 10 into x to the power 9 minus 10 into x plus 1. This is f dash x. This completes the session hope you have understood the concept of derivatives.