Question

Prove from first principles that the function f(x,y) = xsin y + y cos L

Answer 1

 f(x, y) = x sin y + y cos L

By first principle, we have

f _x (x, y) = lim _h$\rightarrow$₀ $\dfrac{f(x + h, y) - f(x, y)}{h}$

f _x (x, y) = lim _h$\rightarrow$₀ $\dfrac{(x + h)sin y + y cosL - (x sin y + y cos L) }{h}$

f _x (x, y) = lim _h$\rightarrow$₀ $\dfrac{(x + h)sin y - x sin y }{h}$

f _x (x, y) = lim _h$\rightarrow$₀ $\dfrac{ h sin y }{h}$

f _x (x, y) = $sin y$

f _y (x, y) = lim _k$\rightarrow$₀ $\dfrac{f(x , y + k) - f(x, y)}{k}$

f _y (x, y) = lim _k$\rightarrow$₀ $\dfrac{x sin (y + k) - (y + k )cosL}{k}$

f _y (x, y) = lim _k$\rightarrow$₀ $\dfrac{x (sin y \cos k + sink\cos y) - (y + k )cosL}{k}$

f _y (x, y) = lim _k$\rightarrow$₀ $\dfrac{x \sin y \cos k }{k} +\dfrac{ sink \ cosy}{k} -\dfrac{ (y + k )cosL}{k}$

On substituting the limit we see that f _y (x, y) does not exist.