Partial Derivatives: Definition, How to Find, Chain Rule, Examples

Derivative is one of mathematics operation especially in calculus. In real life it uses to determine the slope, maximum or minimum of something (income, profit, etc.).

What is Partial Derivative?

Partial derivative is one of way to solve derivative specific problem. In detail, the problem is more than one variable function.

In other words, if there is a function that has x and y as the variables, it will determine the partial derivative with respect of x and another variable (y) is constant and vice versa.

How to Find Partial Derivative?

Finding or determine the value of partial derivative is like determine the value of derivative in general. The different is the function that has more than one variable.

But finding partial derivative is operated the variable one by one or depend on which one that is asked in the problem.

f(x,y) = axm + byn


f’x(x,y) = (a.m)xm-1

f’y(x,y) = (b.n)yn-1


  • x, y        : variables
  • a, b       :coefficient of the x and y
  • m, n     : power of variables
  • f’x(x,y) : partial derivative with respect to x (¶f/¶x)
  • f’y(x,y) : partial derivative with respect to y (¶f/¶y)

Chain Rule Partial Derivatives

Chain rule is one of method to solve complex derivative problem. It is not only use in one variable function problem in derivative but also in more than one variable (partial function).

Remember the concept of chain rule in derivative general problem. If h(x) = (fog)(x) then h’(x) =f’(g(x)) . g’(x).

Chain rule partial derivative concept is not too different.

  • If function has two variables

If h = f(x,y), x = x(t), y = y(t) and h will be operated partial derivative with respect to t, then

dh / dt = (dh/dx) . (dx/dt) + (dh/dy) . (dy/dt)

  • If function has more than two variables.

If h = f(x1, x2, x3, …, xn), for i = 1, 2, 3, …, n, xi = x(t1, t2, …, tn),  and h will be operated partial derivative with respect to t, then

dh / dt = (dh/dx1) . (dx1/dt1) + …. + (dh/dxn) . (dxn/dti)


1. What is partial derivative with respect to x if the function is:

a. f(x) = 2x3 + 4y2 – 1

b. f(x) = 2y – x – 3

c. f(x) = x6 – y2 + x3 – y + 7


a. f’x(x,y) = 6x2

b. f’x(x,y) = -1

c. f’x(x,y) =  6x5 + 3x2

2. If f(x,y) = y2sin2x + x2.ln(y) then determine the partial derivative with respect to y.


remember that x is constant.

f’y(x,y) = 2y.sin2x + (x2/y) = 2y.sin2x + x2y-1

3. Determine partial function with respect to s and t, if the function is

w = yz + zx + xy, x = s2 − t2 , y = s2 + t2 and z = s2.t2.


based on the concept, then

  • dw / ds = (dw/dx) . (dx/ds) + (dw/dy) . (dy/ds) + (dw/dz) . (dz/ds)

              = (z + y)(2s) + (z + x).2s + (x + y).2s.t2

  • dw / dt = (dw/dx) . (dx/dt) + (dw/dy) . (dy/dt) + (dw/dz) . (dz/dt)

             = −(z + y)(2t) + (z + x).2t + (x + y).2s2.t

Read Also

Common Derivatives


Integration by Parts

Integration by Substitution (Reverse Chain Rule)