Question

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

Answer 1

To find the derivative of the given expression we will use chain rule of differentiation. We know that the differentiation of the function $f\left( g\left( x \right) \right)$ is given by $f'\left( g\left( x \right) \right)g'\left( x \right)$ . Hence using this formula we can easily find the solution to the given problem as we know the differentiation of ${{e}^{x}}$ is nothing but ${{e}^{x}}$ and the differentiation of $ax+b$ is a. Also we know that $\dfrac{d\left( cf\left( x \right) \right)}{dx}=c.\dfrac{df\left( x \right)}{dx}$ and hence use this property to find the solution of the given equation.

Complete step-by-step solution:
Now to find the derivative of the given expression we will use chain rule of differentiation.
Chain rule of differentiation helps in differentiating composite function.
Now consider the function $f\left( g\left( x \right) \right)$
Now according to chain rule the differentiation of the function $f\left( g\left( x \right) \right)$ is given by $f'\left( g\left( x \right) \right).g'\left( x \right)$
Now consider the function $f\left( x \right)=4{{e}^{3x+2}}$ .
Now even this is a composite function where $f\left( x \right)={{e}^{x}}$ and $g\left( x \right)=3x+2$
Now we know that $f'\left( x \right)={{e}^{x}}$ hence $f'\left( g\left( x \right) \right)={{e}^{3x+2}}$ and $g'\left( x \right)=3$
Differentiation of ${{e}^{3x+2}}$ by chain rule is ${{e}^{3x+2}}.3....................\left( 1 \right)$
Now consider the differentiation of $4{{e}^{3x+2}}$
Since 4 is a scalar and we know that if c is scalar then differentiation of $cf\left( x \right)$ is given by c × differentiation of $f\left( x \right)$ Hence we have the property $\dfrac{d\left( cf\left( x \right) \right)}{dx}=c.\dfrac{df\left( x \right)}{dx}$ . Using this we get,
$\dfrac{d\left( 4{{e}^{3x+2}} \right)}{dx}=4\dfrac{d\left( {{e}^{3x+2}} \right)}{dx}$
Now substituting the value from equation (1) we get,
$\begin{aligned} & \Rightarrow \dfrac{d\left( 4{{e}^{3x+2}} \right)}{dx}=4{{e}^{3x+2}}.3 \\\ & \Rightarrow \dfrac{d\left( 4{{e}^{3x+2}} \right)}{dx}=12{{e}^{3x+2}} \\\ \end{aligned}$
Hence the differentiation of the given equation is $12{{e}^{3x+2}}$.

Note: Note that while we can use this rule for a long chain of composite functions.
For example the differentiation of $f\left( g\left( h\left( x \right) \right) \right)$ is $f'\left( g\left( h\left( x \right) \right) \right).g'\left( h\left( x \right) \right).h'\left( x \right)$ . Similarly we can do for higher chain of composite functions. Also note that in the formula we have $f'\left( g\left( x \right) \right)$ and not $f'\left( g'\left( x \right) \right)$.