Математика. Материалы курса

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Our practical session will be devoted to the calculation of partial derivatives.

To solve the tasks we are to be able to differentiate the function of one variable, to know the table of derivatives and rules of differentiation.

Thus, the first task, find the partial derivative f_х’(1,2) of function f(x,y)=x²+2xy-y by definition.

The function derivative at point x₀ (by definition) is a ratio limit of the private increment to variable x of function f (x₀) on the argument increment Δх, at Δх->0.

We are to find limit (Δх f(1,2)) / Δх, at Δх->0 in the task

We calculate at first numerator Δхf(1,2). 

Partial increment is the change in the function value when changing only one variable x.

During verbal proof we obtain , Δхf(1,2)=f(1+Δх) - f(1,2)=(1+ Δх)2+4(1+ Δх)-5= Δх(6+Δх).

After substituting converted expression into the required limit we see that when Δх->0,  the limit is 6.

Let’s check if the problem is solved correctly. The partial derivative f_х’(х,у)=2х+2у, f_х’(1,2)=6.

The solution is correct.

It is clear that when solving problems to find a partial derivative, calculation of limits is not the most effective way of solving, it is better to use more quick and convenient techniques

We now turn to solving the following task. We’ll find all partial derivatives of the first order in function of z=x^y^x,то есть z_x’, z_y’.

Let's start with the derivative z_x’.

Indicator here is у^х, we note that variable x is found both in the base and in the power index.

To find the derivative we use a method of differentiation of power-exponential function (see video).

Let’s deal with derivative z_у’.

The variable y is only found in the index, therefore, is the exponential function.

Note that the z - is a complex function, so do not forget to multiply the result by the derivative у^х at y, to calculate we apply rules of differentiation of power function (see video)

Let’s find second-order partial derivatives of function z = x ^ y.

On the first stage it is quite simple, z_x’=ух^(у-1), z_y’= x^y*ln x.

So, to find second-order partial derivatives, we have to find four derivatives z_xх’’, z_xу’’, z_ух’’, z_уу’’.

We get two pure partial derivatives and two mixed ones.

If we compare the answers, we notice that mixed derivatives are equal, it is confirmed by the findings of the theorem of mixed derivatives equation.

Now let’s find all the third order partial derivatives of function z=sin(2x-3y).

Let us make sure that from all third-order derivatives of the proposed function, it is possible to distinguish four different ones.

So, to solve the problem it is enough to find two pure derivatives  z_xхх’’’ and  z_ууу’’’ and two mixed derivatives z_xху’’’ and z_уух’’’ (see video).

All tasks are solved.