 Hi and welcome to the session. Let's work out the following question. The question says find the derivative of e raised to power root x with respect to x from first principle. Let us start with the solution to this question. We have y is equal to e raised to power square root x. We call this 1. So y plus del y will be equal to e raised to power square root x plus del x and we call this equation 2. This implies del y is equal to e raised to power x plus del x minus y that is e raised to power square root x. This implies del y is equal to e raised to power square root x into e raised to power square root x plus del x minus square root x minus 1 divided by square root x plus del x minus square root x and this multiplied by square root x plus del x minus square root x. Now dividing by del x we get del y by del x is equal to e raised to power square root x. Also taking limit del x approaching to 0 we have e raised to power square root x plus del x minus square root x minus 1 divided by square root x plus del x minus square root x multiplied by square root x plus del x minus square root x. This is equal to e raised to power square root x. Now we see that limit del x approaching to 0 e raised to power square root x plus del x minus square root x minus 1 divided by square root x plus del x minus square root x. This limit will be 1. This happens because we see that limit x approaching to 0 e raised to power x minus 1 divided by x is equal to 1. Now here if del x approaches to 0 then square root x plus del x minus square root x also approaches to 0 so we can apply this formula this multiplied by limit del x approaching to 0. This can be written as square root x plus del x minus square root x multiplied by square root x plus del x plus square root x divided by square root x plus del x plus square root x. Here we have rationalized this this is equal to e raised to power square root x multiplied by 1 into this is limit del x approaching to 0. Here we can apply the formula a minus b into a plus b is equal to a square minus b square so we have x plus del x minus x is divided by square root x plus del x plus square root x. This is equal to e raised to power square root x into 1 into this gets cancelled with this. Now here also we see that in all these steps, this is all divided by del x here also we will have a del x here also we will have a del x. Now we see that this del x gets cancelled with this del x so we have limit del x approaching to 0 1 upon square root x plus del x plus square root x we put del x to be equal to 0 and we get e raised to power square root x divided by 2 root x. So our answer to this question is e raised to power square root x divided by 2 root x. So I hope that you understood the solution and enjoyed the session. Have a good day.