Question

Differentiation of ${\left( {2x + 3} \right)^6}$ with respect to $x$ is
A. $12{\left( {2x + 3} \right)^5}$
B. $6{\left( {2x + 3} \right)^5}$
C. $3{\left( {2x + 3} \right)^5}$
D. $6{\left( {2x + 3} \right)^6}$

Answer 1

As the given equation of the form ${\left( {ax + b} \right)^n}$ , in which $x$ is an unknown term and a function is said to be differentiable if its derivative exists and here, we need to find the derivative with respect to $x$, hence differentiate the terms with $\dfrac{d}{{dx}}$ of each term as given in the equation.

Complete step by step answer:
We need to find the derivative with respect to $x$. The equation given is of the form ${\left( {ax + b} \right)^n}$ in which to calculate the derivative of a sum, we simply take the sum of the derivatives. Let us write the given equation
${\left( {2x + 3} \right)^6}$
To find its derivative with respect to $x$ let us write it in simplified manner
$\dfrac{d}{{dx}}\left[ {{{\left( {2x + 3} \right)}^6}} \right]$

Further simplifying the terms, we get
$6{\left( {2x + 3} \right)^5} \cdot \dfrac{d}{{dx}}\left[ {2x + 3} \right]$
Now let us expand the above terms with respect to $x$
$6{\left( {2x + 3} \right)^5}\left( {2 \cdot \dfrac{d}{{dx}}\left[ x \right] + \dfrac{d}{{dx}}\left[ 3 \right]} \right)$
On further simplifying the terms and arranging the terms we get
$6{\left( {2x + 3} \right)^5}\left( {2 \cdot 1 + 0} \right)$
Hence the differentiation of given equation with respect to $x$ is
$12{\left( {2x + 3} \right)^5}$

Therefore, option $A$ is the right answer for this question.

Note: To find any type of derivative with respect to $x$ take $\dfrac{d}{{dx}}$ and with respect to $y$ take $\dfrac{d}{{dy}}$. It is based on the derivative terms asked in the equation.We can find this sum using chain rule to find derivatives with respect to the terms asked in the equation as the rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary. The formula of chain rule is
$f\left( x \right) = \left( {g \cdot h} \right)\left( x \right)$
Therefore, we get
$f\left( x \right) = g\left[ {h\left( x \right)} \right]$
Then
$f'\left( x \right) = g'\left[ {h\left( x \right)} \right] \cdot h'\left( x \right)$
Note that because two functions, $g$and $h$, make up the composite function $f$, you have to consider the derivatives g′ and h′ in differentiating $f\left( x \right)$.