 Hello and how are you all today? The question says find the derivative of sin x square with respect to x from first principle. Let us start the solution. Here let y be equal to sin x square and let change in x be small increment in x whereas change in y be small increment in y, right? So we can write that y plus change in y is equal to sin x plus change in x the whole square. Let this be the second equation and let this be the first equation. Now on subtracting the first equation from the second we get change in y is equal to sin x plus change in x the whole square minus sin x the whole square. This further implies change in y now here we can write it as this can be written as 2 cos plus b by 2 a is x plus change in x the whole square minus sorry plus x square upon 2 sin x plus change in x the whole square minus x square upon 2. This further can be solved and written as change in y is equal to 2 cos bracket x plus change in x the whole square plus x square upon 2 sin sin x change in x plus change in x the whole square upon 2. Now on dividing both the sides by change in x we get change in y upon change in x equal to 2 cos x plus change in x the whole square plus x square upon 2 to sin 2x change in x plus change in x the whole square upon 2. Now this can be written as change in x we can write it in this form also that will be if we divide and then simultaneously multiply 2x change in x plus change in x the whole square upon 2 and hence we will multiply it by 2x change in x plus change in x the whole square upon 2 the whole upon change in x. Now taking limits that is limit change in x approaches to 0 on both the sides now this will be changed into dy by dx equal to limit change in x approaches to 0 2 cos x plus change in x the whole square plus x square upon 2 into limit change in x approaches to 0 sin 2x plus change in x sorry 2x into change in x plus change in x the whole square upon 2 the whole upon 2x change in x plus change in x the whole square upon 2 into limit change in x approaches to 0 2x plus change in x upon 2 into change in x so now we have after using the limits dy by dx equal to 2 cos x plus 0 the whole square plus x square upon 2 this will be equal to 1 into 2x plus 0 upon 2 that on simplifying further will be dy by dx equal to 2x here x will be the remaining thing 2x cos x square as x square plus x square will be 2x square and on dividing 2x square by 2 we will get x square right so this is the required answer to the question given hope you understood it have a nice day