Математика. Материалы курса

Rules of differentiation

Today's lesson is devoted to the differentiation rules training. What do we need to know to prepare for this lesson? The table of derivatives, the formula of derivatives, sums, differences, products, quotient of two functions, the rules for differentiating compositions.

The first example is the simplest. So, the sum and difference of functions are involved here. We apply the rules for differentiating the sum of the difference, and the fact that we can place a constant multiplier out of the sign of the derivative. We wrote the root as x^1/2, and we also wrote the last summands: 1/x is replaced with x^-1 for easy differentiation. 

Next, all we need is to know the formula for the derivative of the power function.

Applying this formula for each of the three cases, we get the following entry. Then we convert, or replace x^-1/2 with 1/root of x, and x^-2 with 1/x^2, we write down the final answer.

Example Two. What question should you always ask yourself?

What is the action performed last in the given formula?

The last action here is division, so we will apply the formula for the derivative of the quotient of these two functions. This is the formula. In the function U and V we know the numerator and the denominator. We get the following entry, and all we have to do is to apply the formulas from the table of derivatives.

Then a small transformation, and we get the final answer.

Example Three. Note, we have a product here, and what is the last action? This is the product of three functions. We had a significant consequence. We discussed it at the lecture. What does it looks like? Note that the derivative of any finite number of the multiplied function can be calculated with the following formula. 

We have 3 functions involved, so there will be 3 summands, in each of which we repeat three functions. We have a stroke in the first summand over the first function; in the second summand it is over the second multiplier, and in the third summand it is over the third function.

Now we have to apply the knowledge of the formulas of the derivatives table. We write it down, and a small transformation leads us to the final answer.

Well, let's now deal with the differentiation of compositions, with the differentiation of a complex function. Again, we should ask: what was the last action performed here? We understand that it was the raising to the 5th power. We have the formula x in the power of alpha; here we have U in the 5th power. We calculate the derivative, and we get 5 U^4 by U’. For this example, we get the following entry.

Now we have to find the derivative 3x-1, obviously, it is three and we get the final answer. 

Another example, the sine of the argument x^2. The last action that was carried out here is the calculation of the sine. The sine of x is a tabular formula. The formula leads us to the following application of the theorem on the derivative of a composite function. The sinus derivative is the cosine of U multiplied by U’. For this example, it is the cosine of the same UX^2 multiplied by x^2. Then the stroke is already above x^2, and we apply the table formula. Then we write down the final answer.

Note, there are also two functions involved here: sine and squaring. However, let's ask a question.

What was the last operation performed for this composition?

It is squaring. But for squaring, the tabular formula U^2 is 2U multiplied by U’, and U is sin x.

Having applied this formula we get 2sin x multiplied by sin'x, this is cos x.

We see that we have obtained the formula for the sine of a double angle. This is the answer.

Now please try to calculate and find the values of these derivatives using the formulas of the differentiation rules, as well as the consequences and the derivative table.