Chainrule I

Chainrule I (math)

The  word chain is used as adjective in many contexts. Something whichhappens not suddenly and directly but in stages is called as chainevent. The vitals of the food we eat is not instantly observed intoour blood stream. It undergoes several processes before that. Henceit is a chain reaction.

Similarlyin calculus, sometimes the derivatives of functions are determined bya chain process. This is because many functions are not directlyrelated to the variable and may some other function of the variable.

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For example, we know the derivative of sin (x) is cos (x) but whatabout the derivative of sin (3x)? Here the function is not related tojust ‘x’ but ‘3x’. Mathematicians found an easy way out tofind the derivatives in such cases and framed a rule for suchmethods. This rule is called as chain rule and the method adopted iscalled as chain differentiation.. More explicitly it is called ascalculus chain-rule as there are rules in other subjects also forchain events.

Aderivative found by the above rule is called as chain rule derivativeor derivative chain rule, meaning the derivative is found by themethod chain differentiation. Let us describe the method of chaindifferentiation.

Let ybe a function of uwhich in turn is a function v of x.As per the rule of chain differentiation, y ‘ = u’ * v’, wherey ‘ is the derivative of ywith respect to x,u’ is the derivative of ywith respect to uand v’ is the derivative of vwith respect to x.One can better understand this rule in Leibniz’s notation (dy/dx) =(dy/du)*(du/dx).

Letus see an example of using chain rule. Suppose a function is definedas y = sin (lnlxl). What is the derivative dy/dx? Let us make asubstitution u = lnlxl and hence the function is decomposed as y =sin (u), where u = lnlxl. So, dy/dx = (dy/du)*(du/dx) = cos (u)*(1/x)= [cos (lnlxl)]/(x)

Therule of chain differentiation not only helps to find derivatives ofcomposite functions but used as a smart technique for easierdifferentiation.

For example, the derivative of a function y = (2x +3)5,can be found by writing the binomial expansion and differentiatingall the addends. But the task is laborious. On the other hand, bychain differentiation, in one stroke we can find the derivative as5(2x + 3)4*2 = 10(2x + 3)4.

Inmany cases of integrations it is easier to make a suitablesubstitution and integrate. Here the same rule is called as chainrule integration.