Partial differentiation is the process of differentiating functions of more than one

independent variable, which in science they usually are. The rule when differentiating with  respect to one variable, is to treat the others as constants, and you can use methods such as the chain and product rules. We can find stationary points, but now we require all of the first order derivates to be zero simultaneously, so we have a set of equations to solve. For a surface given by z =f(x,y), we can test if it is a saddle point, or a minimum/maximum.  If it is a minimum or maximum, we look to the sign of the two second order derivatives to

determine which. Another difference for multiple dimensions is that the gradient or rate of   change of a function depends on the direction we “look in”. This is given by the “directional derivative”, and the direction of  maximum increase is given by the vector ∇ f (“grad f”).

1)  Warm up: Differentiate the following functions with respect to x, (where k is a constant)

i)  f(x) = kekx sin(kx)

ii) f(x) = xekx sin(kx)

∂f        ∂f

2)   For the following, find       and       ,

∂x       ∂y

i)   f(x,y) = xexy sin(xy)

ii)  f(x,y) = yexy sin(x2y)

iii) f(x,y) = cos(sin)

3)   For the  function f(x,y) = x2 +xy2 +x sin(y) +x sin(xy), determine

∂f         ∂f             ∂ 2f             ∂ 2f              ∂ 2f           ∂ 2f

i)        and        ,        ii)         ,         iii)        ,         iv)            and            

∂x         ∂y            ∂x2          ∂y2          ∂x∂y         ∂y∂x

4)   Find the stationary point(s) of the function:

f(x,y) = 2x3 + 6xy2 − 3y3 − 150x + 7.

Remember that a stationary point is one where all (both) of the first derivatives are

zero simultaneously. Hint: One equation can be factorised. Find where each factor can be zero and solve the other equation in each case. There are four stationary points.

Then, with the coordinates (x,y) of each stationary point, determine their nature.

5)  A function in three-dimensional space is given by f(x,y,z) = x2y + 2xyz.

i)  Find  ∇ f.

ii) What is the gradient of f at a point (2,3,3) in the direction of u =  + 2j(̂) −  ?