Question

How do you find the derivative of ${\left( {4x + 1} \right)^2}{\left( {1 - x} \right)^3}$?

Answer 1

When two functions are multiplied together and we have to find the derivative, then we have done the change derivative which means that derivative of the first function plus the derivative of the second function.
Derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument.
By application of chain rule will differentiate the given function.

Complete step-by-step solution:
Let’s discuss the chain rule first and then we will perform the derivative of the function given in the question.
Chain rule is generally computed for the derivative of the composite function.
Let f(x) and g(x) are two composite functions given, and we have to find the derivative of the composite function;
F’(x) g(x) + f(x) g’(x), we performed the differentiation of the first function leaving the second function as it is and we performed the differentiation of the second function leaving the first function as it is.
Now, we will find the derivative of the function given.
$\Rightarrow {\left( {4x + 1} \right)^2}{\left( {1 - x} \right)^3}$(Given function)
$\Rightarrow {(1 - x)^3}\dfrac{{d{{(4x + 1)}^2}}}{{dx}} + {(4x + 1)^2}\dfrac{{d{{(1 - x)}^3}}}{{dx}}$ (We have performed the derivative)
$\Rightarrow 8{\left( {1 - x} \right)^3}(4x + 1) - 3(4x + 1){(1 - x)^2}$ (We have come out with derivative result)
On further simplification;
$\Rightarrow {\left( {1 - x} \right)^2}\left( {4x + 1} \right)\\{ 8(1 - x) - 3\\}$ (We have taken out the common terms out of the bracket)
Using the identity of (a + b)2 we will further simplify the answer
$\Rightarrow - 32{x^4} + 76{x^3} - 43{x^2} + 2x + 5$

Therefore the derivative of the given function is $- 32{x^4} + 76{x^3} - 43{x^2} + 2x + 5$.

Note: Derivatives has many applications such as finding the critical points in mathematics, intervals of increase and decrease, relative maxima and minima, concavity and inflection points, curve sketching with derivative, optimization using the closed intervals etc