Question

How do you find the derivative of \[{e^{\sin x}}\] ?

Answer 1

Hint: In this question, we have to find the derivative of a function that involves e raised to some power so we have to differentiate \[{e^{\sin x}}\] with respect to x. We will first differentiate the whole quantity \[{e^{\sin x}}\] and then differentiate the quantity that is written in the power as it is also a function of x $(\sin x)$ . The result of multiplying these two differentiated functions will give the value of $\dfrac{{dy}}{{dx}}$ or $y'(x)$ .On solving the given question using the above information, we will get the correct answer.

Complete step-by-step solution:
We have to differentiate \[{e^{\sin x}}\]
Let $y = {e^{\sin x}}$
We know that $\dfrac{{d{e^x}}}{{dx}} = {e^x}$
So differentiating both sides of the above equation with respect to x, we get –
$\dfrac{{dy}}{{dx}} = {e^{\sin x}}\dfrac{{d\sin x}}{{dx}}$
We also know that $\dfrac{{d\sin x}}{{dx}} = \cos x$ , so we get –
$\dfrac{{dy}}{{dx}} = \cos x{e^{\sin x}}$
Hence, the derivative of \[{e^{\sin x}}\] is $\cos x{e^{\sin x}}$ .

Note: Differentiation is used when we have to find the instantaneous rate of change of a quantity, it is represented as $\dfrac{{dy}}{{dx}}$ , in the expression $\dfrac{{dy}}{{dx}}$ , a very small change in quantity is represented by $dy$ and the small change in the quantity with respect to which the given quantity is changing is represented by $dx$ .

In this question, we have to differentiate \[{e^{\sin x}}\] , it is a function containing only one variable quantity, so we can simply start differentiating it. But if the equation contains more than one variable quantity, then we must rearrange the equation first for solving similar questions so that the variable with respect to which the function is differentiated is present on one side and the variable whose derivative we have to find is present on the other side.