 Hello and welcome to the session. Let's discuss the following question. It says differentiate e to the power sin x plus sin x to the power x with respect to x. So let's now move on to the solution and let y be the function e to the power sin x plus tan x to the power x. Let u be equal to e to the power sin x and v be equal to tan x to the power x. Therefore y is equal to u plus v and dy by dx is equal to du by dx plus dv by dx. Now u is e to the power sin x taking log on both sides we have log u is equal to log e to the power sin x which is equal to sin x log e which is equal to sin x. So we have log u is equal to sin x. Now we differentiate both sides with respect to x. Now the derivative of u with respect to x is derivative of log u is 1 by u du by dx is equal to cos x derivative of sin x is cos x. So this implies du by dx is equal to u cos x. So this implies du by dx is equal to u is e to the power sin x that is e to the power sin x into cos x. Now v is tan x to the power x. So log v is equal to x log tan x. So now we differentiate both sides with respect to x so we have derivative of log v with respect to x is 1 by v dv by dx. Now to differentiate this we need to use the product rule. So we have x into derivative of log tan x which is 1 upon tan x into derivative of tan x which is secant square x plus log tan x into derivative of x with respect to x is 1. So this is equal to 1 upon v dv by dx is equal to x into secant square x upon tan x plus log tan x. So this implies dv by dx is equal to v into x secant square x can be written as 1 upon cos square x upon tan x tan x is written as sin x upon cos x plus log tan x. Now v is tan x to the power x. Now here cos x it's cancelled with cos x and we have x upon cos x into sin x plus log tan x. Now dv by dx is equal to dv by dx which is this and dv by dx which is this. So we have dv by dx is e to the power sin x into cos x plus dv by dx is tan x to the power x into x upon cos x into sin x plus log tan x. So we have differentiated the given function and this is the required answer. So this completes the question and the session. Bye for now. Take care. Have a good day.