Find expressions for the partial derivatives of the following functions:

a. f (x, y) = f (g (x) k (y), g (x) + h (y)

b. f (x, y, z) = f (g ( + y), h (y + z))

c. f (x, y, z) = f (xy, yz, zx)

d. f (x, y) = f (x, g (x), h (x, y))

Step-by-step explanation: partial derivative is the differentiation of one variable e. g. X while leaving the values of the other variable e. g. Y

These four questions A, B, C and D have different functions separated by commas. I will not assume the commas to be something else like a plus sign.

A. f (x) = g' (x). k (y), g' (x) + h (y)

f (y) = k' (y). g (x), g (x) + h' (y)

B. f (x) = g'x (x+y)

f (y) = g'y (x+y), h'y (y+z)

f (z) = h'z (y+z)

C. f (x) = f'x (xy), f'x (zx)

f (y) = f'y (xy), f'y (yz)

f (z) = f'z (yz), f'z (zx)

D. f (x) = f'x (x), g' (x), h'x (x, y)

f (y) = h'x (x, y)

These are the partial derivative expressions for each variable in each function. You will need to pay a lot of attention to understand:

* while differentiating X alone, functions in Y which are separated by commas from the functions in X, are ignored totally because they are different questions

* In functions where X added to Y is in a bracket e. g. (x+y), to find the derivative of X, Y isn't thrown away because they are joined (by a plus sign) the derivative of X alone in this case would be f'x (x+y)

* f (x), just like g (x), simply means/represents a function in X hence f' (x) means the differentiation of all X-terms in that function