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Question: How do you know whether to use chain, product or quotient rule?...

How do you know whether to use chain, product or quotient rule?

Explanation

Solution

Hint : The rules mentioned above are the rules related to differentiation. These are generally used when two functions are there in the differentiation. The only thing is the way they used and the place where they used are different. We are going to get the knowledge of where and how they are used.

Complete step by step solution:
I.Product rule:
If f and g are the two differentiable functions then the derivative of their product will be given by the product rule. That is mentioned below;
ddx[f(x)g(x)]=f(x)g′(x)+g(x)f′(x)\dfrac{d}{{dx}}\left[ {f\left( x \right)g\left( x \right)} \right] = f\left( x \right)g'\left( x \right) + g\left( x \right)f'\left( x \right)
This is the sum of two different terms that are in a way “first term as it is multiplied by the derivative of the second and it is added with the second term as it is multiplied by the derivative of the first”.

II.Chain rule:
If f and g are the two differentiable functions then the derivative of their chain will be given by the chain rule. That is mentioned below;
ddx[f(g(x))]=f′(g(x))g′(x)\dfrac{d}{{dx}}\left[ {f\left( {g\left( x \right)} \right)} \right] = f'\left( {g\left( x \right)} \right)g'\left( x \right)
It is like the derivative of the first function inculcated in the second.

III.Quotient rule:
If f and g are the two differentiable functions then the derivative of their quotient will be given by the quotient rule. That is mentioned below;
ddx[f(x)g(x)]=g(x)f′(x)−f(x)g′(x)(g(x))2\dfrac{d}{{dx}}\left[ {\dfrac{{f\left( x \right)}}{{g\left( x \right)}}} \right] = \dfrac{{g\left( x \right)f'\left( x \right) - f\left( x \right)g'\left( x \right)}}{{{{\left( {g\left( x \right)} \right)}^2}}}
Here in this case the numerator is the same as the product rule but the denominator is the square of the function in the denominator of the original function.

Note : Note that the derivative of sum and difference of two functions results in the sum and difference of their derivatives only. But this is not in case of product and division. That’s why these rules are used. Also note that the chain rule is like two embedded functions and product rule has two separate functions.