Question

If the derivative of $\left( {ax - 5} \right){e^{3x}}$ at $x = 0$ is $- 13$, then the value of $a$is equal to

$$(A){\text{ 8}} \\\ (B){\text{ - 5}} \\\ (C){\text{ 5}} \\\ (D){\text{ - 2}} \\\ (E){\text{ 2}} \\\$$

Answer 1

Hint:- Use the product rule to find derivatives.

Let, y$= \left( {ax - 5} \right){e^{3x}}$ (1)
As given in the question that the value of derivative of y with respect to x at $x = 0$is $- 13$.
As, y is a function of x. So, we can get the derivative of y easily by using product rule.
Which states that if u and v are two functions then,
$\Rightarrow \left( {\dfrac{{d(uv)}}{{dx}}} \right) = u\dfrac{{dv}}{{dx}} + v\dfrac{{du}}{{dx}}$
So, here u$= {e^{3x}}$ and v$= \left( {ax - 5} \right)$
So, differentiating equation 1 with respect to x. We get,
$\Rightarrow \dfrac{{dy}}{{dx}} = {e^{3x}}\dfrac{{d(ax - 5)}}{{dx}} + (ax - 5)\dfrac{{d({e^{3x}})}}{{dx}}$ (By using product rule)
$\Rightarrow \dfrac{{dy}}{{dx}} = {e^{3x}}(a) + (ax - 5)3{e^{3x}}$
Now, putting $x = 0$ in the above equation. We get,
$\Rightarrow {\left( {\dfrac{{dy}}{{dx}}} \right)_{x = 0}} = {e^0}(a) + (a(0) - 5)3{e^0}$
As, given in the question, the derivative of the given function i.e. y at $x = 0$ is $- 13$. So,
$\Rightarrow - 13 = a - 15$
$\Rightarrow a = 2$
Hence, the correct option is E.

Note:- Whenever we come up with this type of problem where we are given with a function and the value of the derivative of that function at a given point, we first calculate the derivative of that function at a known point, then equate it with the given value of the derivative of the function at that point to get the required value of the variable.