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A derivative of a function is the rate of change of one variable involved in the function, say x with respect to corresponding change of another variable in the same function, say y. The method we use to find derivatives is called differentiation. We use differential rules to solve the questions more quickly and efficiently instead of using the more time occupying differentiation method involving limits. These are general rules that are very commonly useful in solving more complex functions.
Derivation
Suppose f is a real valued function and a is a point in its domain of definition. The derivative of f at a is defined by
It should be provided that this limit exists before attempting derivation. Derivative of f (x) at a is denoted by f′(a). f’(a) is the change in f(x) at a with respect to x.
This is the standard method used for calculating differentiation but it is suitable for simpler functions only. We define these rules to make solving complex functions easier.
Differential Rules
Differential rules can be proved all by using the above method of differentiation. The four basic differential rules deal with the basic mathematical operations of addition, subtraction, multiplication and division and how they manifest in the algebra of derivatives of functions. There is also a rule to deal with the power of a function and also one for reciprocal of a function. The most important of all is the chain rule that we use to deal with composition of two functions and to find derivatives for that composition.
Sum Rule
Derivative of the sum of two functions is the sum of the derivatives of the functions.
It can be written as (u+v)’= u’+v’.
Difference Rule
Derivative of the difference of two functions is the difference of the derivatives of the functions.
It can also be denoted like (u-v)’=u’-v’
Product Rule
Derivative of the product of two functions is given by the following product rule.
Another form: (uv)’=u’v+uv’
Quotient Rule
Derivative of two functions getting divided is given by the following quotient rule (whenever the denominator is non–zero). This rule is less straightforward than the others.
Generally written as, (u/v)’ = (u’v-uv’)/v²
We can derive the quotient rule from the derivation method as follows:
Let there be two functions f and g. To find the derivative of f/g,
Multiplication by constant
When a constant is multiplied with a function. The derivative of it comes out to be equivalent to multiplication of constant and derivation of that function individually. Basically the constant brings no change to the derivation of the function except getting multiplied with the derivative to give the final result.
(cf)’ = cf’ where c is the constant and f is a function.
Power Rule
It states that a function containing an integer as a power say n will have a derivative with a constant n and power raised to the function would become one less than what was before.
f(x) = xn is nxn-1 for any positive integer n.
Reciprocal Rule
When we apply derivation on a reciprocal of a function then the derivative is negative one by function square times.
(1/f)’ = -f/f².
Chain Rule
Chain rule is used to calculate the derivation of the composite of two functions. It is one of the most important rules in derivation as highly complex functions can be solved using this.
It states that the derivative of composite of two functions f and g result in the derivation,
(fog)’ = f’(g(x))g’(x).
Conclusion
Differential rules are general rules in differentiation that are very commonly useful in solving more complex functions. We can derive all the differential rules from the derivation method. In questions, they are used directly. The four basic differential rules deal with the basic mathematical operations of addition, subtraction, multiplication and division and their usage in algebra derivatives of functions. Multiplication by constant, power rule, reciprocal rule and chain rule are some of the other more important differential rules.
