Question

If \[ f(x,y,z,w)=x^{2}e^{2y+3z}\cos(4w), \] then \[ \frac{\partial f}{\partial z} \] at \((2,0,-2,1)\) is

• A. \[ \frac{8\cos4}{e^{4}} \]
• B. \[ \frac{12\cos4}{e^{6}} \]
• C. \[ \cos4 \]
• D. \[ e \]

Hint:

For functions of the form \[ f=x^m e^{ax+by+cz}\sin(kz) \quad\text{or}\quad f=x^m e^{ax+by+cz}\cos(kz), \] while taking the partial derivative with respect to one variable:
• Treat all other variables as constants.
• Apply the chain rule only to the exponential term.
• Substitute the given point only after obtaining the derivative. This saves time and reduces calculation errors in competitive examinations.

Answer 1

The Correct Option is B