Question

How do you find the derivative of $x{(x - 4)^3}$?

Answer 1

Hint: According to give in the question we have to determine the derivation of the given function which is $x{(x - 4)^3}$ so, to find the derivation of the function as we can see that there are two functions to which we have to differentiate and to differentiate them we have to follow the formula which is as explained below:

Formula used: $ \Rightarrow \dfrac{d}{{dx}}uv = u\dfrac{d}{{dx}}v + v\dfrac{d}{{dx}}u...................(A)$
According to the formula (A) above, u is the first function and v is the second function we can also take v as the first function and u as the second function so, we have to take the first term as constant term and then differentiate the second term we choose and then we have to add it with the second term as the constant term and differentiation of the first term.
Now, to find the differentiation we have to use the formulas which are as mentioned below:
$ \Rightarrow \dfrac{d}{{dx}}x = 1.....................(B)$
$ \Rightarrow \dfrac{d}{{dx}}{(x - a)^n} = n{(x - a)^{n - 1}}................(C)$

Complete step-by-step solution:
Step 1: First of all we have to take our first term and second term in this case I will choose x as a first term and ${(x - 4)^3}$ as a second term we can also choose ${(x - 4)^3}$as a first term and x as a second term.
Step 2: Now, to find the differentiation of the function we have to use the formula (A) which is as mentioned in the solution hint. Hence,
$
   \Rightarrow \dfrac{d}{{dx}}x{(x - 4)^3} = x\dfrac{d}{{dx}}{(x - 4)^3} + {(x - 4)^3}\dfrac{d}{{dx}}x.............(1) \\
    \\
 $
Step 3: Now, to solve the expression (1) as obtained in the solution step 2 we have to use the formula (B) to find the differentiation of x. Hence,
$
   \Rightarrow \dfrac{d}{{dx}}x{(x - 4)^3} = x\dfrac{d}{{dx}}{(x - 4)^3} + {(x - 4)^3} \times 1.............(2) \\
    \\
 $
Step 4: Now, to solve the expression (2) as obtained in the solution step 3we have to use the formula (C) to find the differentiation of ${(x - 4)^3}$. Hence,
\[
   \Rightarrow \dfrac{d}{{dx}}x{(x - 4)^3} = 3x{(x - 4)^{3 - 1}} + {(x - 4)^3} \\
   \Rightarrow \dfrac{d}{{dx}}x{(x - 4)^3} = 3x{(x - 4)^2} + {(x - 4)^3}..........(3)
 \]
Step 5: Now, from the expression (3) as obtained in the solution step 4 we have to take the term \[{(x - 4)^2}\] to solve the expression. Hence,
\[
   \Rightarrow \dfrac{d}{{dx}}x{(x - 4)^3} = {(x - 4)^2}\{ 3x + (x - 4)\} \\
   \Rightarrow \dfrac{d}{{dx}}x{(x - 4)^3} = 4{(x - 4)^2}(x - 1)
 \]

Hence, with the help of the formulas (A), (B), and (C) we have determined the differentiation of the given function $x{(x - 4)^3}$ is \[\dfrac{d}{{dx}}x{(x - 4)^3} = 4{(x - 4)^2}(x - 1)\].

Note: If there are two terms in the function to be differentiated then we have to choose the first and the second term and then we have to take the first term as constant term and differentiate the second term and then we have to take the second term as constant term and differentiate the first term.
It is necessary that we have to add both of the differentiation we found after choosing the first and second term and their differentiation.