How to find partial derivatives in three or more variables

When a function has several inputs, a partial derivative tells us what changes when we vary just one of them. We’ll work through a function of three variables first, then use the same idea with four.

Differentiate with respect to one variable at a time

The notation fx or ∂f∂x means we differentiate with respect to x while holding the other inputs constant. The ordinary derivative rules still apply to the variable we chose.

Find the three partials of a product

f(x,y,z)=3x5y2z3

For fx, the factor 3y2z3 stays fixed, so we differentiate x5. For fy, the factor 3x5z3 stays fixed. For fz, differentiate z3 while keeping 3x5y2 fixed.

fx=3(5x4)y2z3=15x4y2z3fy=3x5(2y)z3=6x5yz3fz=3x5y2(3z2)=9x5y2z2

Find all four partials when terms use different variables

f(w,x,y,z)=2wx4-3y37z2+sin(2x)

Only the first term involves w. Two terms involve x, so for fx use the power rule on x4 and the chain rule on sin(2x).

fw=2x4fx=8wx3+2cos(2x)

The middle term contains both y and z. For fy, hold z fixed and differentiate y3. For fz, hold y fixed and write the term as -3y37z-2.

fy=-9y27z2fz=-3y37(-2)z-3fz=6y37z3

The function and these partial derivatives are defined for z≠0.

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