If f(x) = , find the value of f'(1).

Concept:

• xn = nxn-1
• sin x = cos x
• cos x = -sin x
• ex = ex
• ln x = 
• (ax + b) = a
• tan x = sec2 x
• f(x)g(x) = f'(x)g(x) + f(x)g'(x)
•  sin-1 x = 
•  tan-1 x = 

Calculation:

Let y = f(x) = 

Taking log both sides, we get

⇒ ln y = ln 

ln y = tan^-1 x (ln x)                      (∵ log mn = n log m)

Differentiating with respect to x, we get

 = 

 = y × 

 = 

f'(x) =  = 

Now f'(1) = 

f'(1) =  or 45°