Q.18 If u = log (ex + ey), then
∂u/∂x + ∂u/∂y =
- ex + ey
- ex − ey
- 1 / (ex + ey)
- 1
Introduction
Partial differentiation is a key concept in calculus and is commonly tested in competitive
examinations such as GATE, IIT-JAM, CSIR-NET, and university exams. In this problem, we evaluate
the sum of partial derivatives of a logarithmic function involving exponential terms.
Given Function
u = log ( ex + ey )
We are required to find:
∂u/∂x + ∂u/∂y
Step-by-Step Solution
Step 1: Find ∂u/∂x
∂u/∂x = ex / ( ex + ey )
Step 2: Find ∂u/∂y
∂u/∂y = ey / ( ex + ey )
Step 3: Add the Partial Derivatives
∂u/∂x + ∂u/∂y = ( ex + ey ) / ( ex + ey )
= 1
Final Answer
Correct Option: (D) 1
Key Takeaways
- Use the chain rule for logarithmic differentiation
- Partial derivatives can simplify significantly after addition
- Symmetry in exponential terms often leads to constant results
