The problem of finding of a derivative faces both pupils of high school of schools, and students. For successful differentiation it is required to conform attentively and accurately to certain rules and algorithms.
It is required to you
- - table of derivatives;
- - rules of differentiation.
Instruction
1. Analyze a derivative. If it represents the work or the sum, spread out by the known rules. In case one of composed — number, use formulas from Paragraphs 2-5 and 7.
2. Remember that the derivative of number (constant) is equal to zero. The derivative by definition is the function change speed, and speed of change of a constant — zero. If necessary it is proved by means of definition of a derivative, through limits — increment of function is equal to zero, and to divide zero into increment of an argument there is zero. Therefore, zero limit are zero too.
3. Do not forget that, having the work of a constant multiplier and variable, it is possible to take out a constant for the sign of a derivative and to differentiate only the remained function: (cU) of '=cU', where "with" – a constant; "U" is any function.
4. Having one of special cases of derivative fraction when in numerator instead of function there is a number, use a formula: the derivative is equal minus to the work of a constant on a denominator derivative, divided into the function standing in a denominator in a square: (with / U) '= (-c · U') / U2.
5. Take a derivative on the second investigation of derivative fraction: if the constant costs in a denominator, and in numerator function, then unit divided into a constant, all the same number therefore it is necessary to take out number from under the sign of a derivative and to change only function: (U/c) =(1/c)·U'.
6. Distinguish coefficient before an argument ("x") and before function (f(x)). If the number faces an argument, then function — difficult, and it needs to be differentiated by rules of difficult functions.
7. If you have an exponential function ah, in this case the number is built in variable degree that is, it is necessary to take a derivative on a formula: (ah) '=lna · ah. Be careful and remember that any positive number other than unit can be the basis of an exponential function. If the basis of an exponential function — number e, then the formula takes a form: (ekh)' = ekh.