Differentiate the following functions with respect to x:

Let, y = e^x log √x tan x = e^x log x^1/2 tan x = 1/2 e^x log x tan x

We have to find dy/dx

As we can observe that y is a product of three functions say u, v & w where,

u = log x

v = e^x

w = tan x

∴ y = 1/2 uvw

As we know that to find the derivative of product of three function we apply product rule of differentiation.

By product rule, we have –

…equation 1

As, u = log x

∴ …equation 2 {∵ }

As, v = e^x

…equation 3 {∵ }

As, w = tan x

∴ …equation 4 {∵ }

∴ from equation 1, we can find dy/dx

∴

using equation 2, 3 & 4,we have –

⇒

Hence,