Question

How do you use the Quotient Rule to prove the Reciprocal Rule?

Answer 1

To solve such questions, a basic knowledge of differentiation rules is required. Write Quotient Rule by taking $f$ and $g$ as the two differentiable functions. Then use the differentiation properties on the Quotient Rule to finally obtain the Reciprocal Rule.

Formulae Used:
The Quotient Rule in differentiation is given as $\dfrac{d}{{dx}}\left( {\dfrac{f}{g}} \right) = \dfrac{{\dfrac{d}{{dx}}(f).g - \dfrac{d}{{dx}}(g).f}}{{{g^2}}}$ where $f$ and $g$ are the two differentiable functions.

Complete step-by-step answer:
We need to prove that
$\dfrac{d}{{dx}}\left( {\dfrac{1}{g}} \right)(x) = \dfrac{{ - g'(x)}}{{{{(g(x))}^2}}}$
Where $g'(x)$ indicates the first derivative of the function $g$ , that is, $g'(x) = \dfrac{d}{{dx}}(g)$ .
Assume that $g$ is differentiable at $x$ and that $g(x) \ne 0$ .
Now, apply the quotient rule to the two functions $f$ and $g$ ,
$\dfrac{d}{{dx}}\left( {\dfrac{{f(x)}}{{g(x)}}} \right) = \dfrac{{f'(x)g(x) - f(x)g'(x)}}{{{{(g(x))}^2}}}$
Where, $f'(x) = \dfrac{d}{{dx}}(f(x))$ and $g'(x) = \dfrac{d}{{dx}}(g(x))$ respectively.
Let the function $f$ be a constant such as $1$ and hence the derivative of the function $f$ will be,
$\dfrac{d}{{dx}}(f(x)) = \dfrac{d}{{dx}}(1) = 0$
Substitute the value of the function $f$ and its derivative $f'(x)$ in the Quotient Rule to get,
$\dfrac{d}{{dx}}\left( {\dfrac{1}{{g(x)}}} \right) = \dfrac{{0.g(x) - 1.g'(x)}}{{{{(g(x))}^2}}}$
Simplify the expression further to get,

**$\Rightarrow \dfrac{d}{{dx}}\left( {\dfrac{1}{{g(x)}}} \right) = \dfrac{{ - g'(x)}}{{{{(g(x))}^2}}}$
Which is the required Reciprocal Rule. Hence, the Reciprocal Rule is proved by using the Quotient Rule. **

Additional Information: The Quotient Rule in calculus is a fairly easy method of finding the derivative of a function that is the ratio of two differentiable functions. The Quotient Rule is easily the most used in calculus after the chain rule and helps to make calculations pretty simple. The Reciprocal Rule in calculus gives the derivative of the reciprocal of a function $f$ in terms of the derivative of $f$ . This rule can be used to deduce the quotient rule as well as the product rule.

Note:
Keep in mind that the denominator of the quotient rule is always squared and is the function that is in the denominator of the ratio to be differentiated. Also, always write the derivatives of both the functions in the ratio separately and then substitute them in the formula to keep the calculations clear and simple.