Question

Find \[\dfrac{dy}{dx}\] of \[y={{\left( \dfrac{x-5}{2x+1} \right)}^{3}}\]

Answer 1

Hint: To solve the above problem we have to know the basic derivatives of \[{{x}^{3}}\] and internal derivatives. After writing the derivatives rewrite the equation with the derivatives of the function. \[\dfrac{d}{dx}{{\left( y \right)}^{n}}=n.{{y}^{n-1}}\dfrac{dy}{dx}\] We can see one function is inside another we have to find internal derivative.

Complete step-by-step answer:
\[y={{\left( \dfrac{x-5}{2x+1} \right)}^{3}}\]. . . . . . . . . . . . . . . . . . . . . (a)
\[\dfrac{d}{dx}{{\left( y \right)}^{n}}=n.{{y}^{n-1}}\dfrac{dy}{dx}\]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)
\[\dfrac{d}{dx}{{\left( x \right)}^{3}}=3{{x}^{2}}\]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)
Substituting (1) and (2) in (a) we get,
Therefore derivative of the given function is,
\[\dfrac{dy}{dx}=3{{\left( \dfrac{x-5}{2x+1} \right)}^{2}}\left( \dfrac{d}{dx}\left( \dfrac{x-5}{2x+1} \right) \right)\]
Now the quotient rule is applied, By writing the derivatives we get,
Further solving we get the derivative of the function as
\[\dfrac{dy}{dx}=3{{\left( \dfrac{x-5}{2x+1} \right)}^{2}}\left( \dfrac{2x+1(\dfrac{d}{dx}\left( x-5 \right)-(x-5)(\dfrac{d}{dx}\left( 2x+1 \right))}{{{(2x+1)}^{2}}} \right)\]
By solving we get,
\[\dfrac{dy}{dx}=3{{\left( \dfrac{x-5}{2x+1} \right)}^{2}}\left( \dfrac{(2x+1)(1)-(x-5)(2)}{{{(2x+1)}^{2}}} \right)\]
\[\dfrac{dy}{dx}=3{{\left( \dfrac{x-5}{2x+1} \right)}^{2}}\left( \dfrac{(2x+1)-(2x-10)}{{{(2x+1)}^{2}}} \right)\]
\[\dfrac{dy}{dx}=3{{\left( \dfrac{x-5}{2x+1} \right)}^{2}}\left( \dfrac{11}{{{(2x+1)}^{2}}} \right)\]

Note: In the above problem we have solved the derivative of a cubic expression and we found out the internal derivative by quotient rule. In this case we have to find an internal derivative. Further solving for \[\dfrac{dy}{dx}\]made us towards a solution. If we are doing derivative means we are finding the slope of a function. Care should be taken while doing calculations.