VIDEO: Calculate the derivative 1 through x

If you want to derive the function "1 through x", then you either have to transform the function or be familiar with the calculation rule.

The derivative of 1 by x

1. In order to be able to form the correct derivative, you must first transform the function.
2. A function of the form 1 through x (1 / x) can be turned into a function of the form x-1 with the help of the power laws.
3. The derivation of the function x-1 is much more straightforward. The general rule of derivation for power functions applies: xn -> n * xn-1. You can also apply this rule to rational exponents.
4. According to this rule, you pull the exponent as a factor in front of the x. Then the exponent is decreased by 1.


Derive 2 by x - this is how it works with fractional-rational functions

If you want to derive the function "2 by x", you can do this with a little ...

5. For the concrete function this would look like this: x-1 -> -1 * x-2.
6. Since 1 can be neglected as a factor, you arrive at the intermediate result - x-2.
7. If you undo the reshaping step that you carried out at the beginning, you will get the following end result for the derivation: - 1 through x2 (-1 / x²).
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8. Now do you want a general rule for Functions with negative exponents, you must first determine another one of this type.
9. As an example, function 1 through x2. Repeat the above steps for this function, then you will get the intermediate result - 2 * x-3.
10. If you now use the reshaping step for this function, you will arrive at this derivation: - 2 / x3.
11. You can use this derivation to identify a scheme. The numerator is replaced by the exponent of x. Then the exponent of x is increased by 1. Finally, a "-" is placed in front of the function.
12. If you want to formulate this in a mathematical way, it would look like this: 1 through xn -> (- n) through xn + 1.
13. If you have higher Derivatives then apply the same steps again.
14. If you want to derive the first derivative, you have to carry out this calculation step: - 1 / x2 = - x-2.
15. After you have applied the reshaping step again, you must now derive: - (- 2) * x-3 = 2 * x-3.
16. If you now undo the transformation, the end result for the second derivative is: 2 / x3.

A general rule

1. If you now want to define a general rule for functions with negative exponents, you must first define another of this type.
2. As an example, function 1 through x2. Repeat the above steps for this function, then you will get the intermediate result - 2 * x-3.
3. If you now use the reshaping step for this function, you will arrive at this derivation: - 2 / x3.
4. You can use this derivation to identify a scheme. The numerator is replaced by the exponent of x. Then the exponent of x is increased by 1. Finally, a "-" is placed in front of the function.
5. If you want to formulate this in a mathematical way, it would look like this: 1 through xn -> (- n) through xn + 1.

Form higher derivatives

1. If you want to take higher derivatives, then apply the same steps again.
2. If you want to derive the first derivative, you have to carry out this calculation step: - 1 / x2 = - x-2.
3. After you have applied the reshaping step again, you must now derive: - (- 2) * x-3 = 2 * x-3.
4. If you now undo the transformation, the end result for the second derivative is: 2 / x3.