Answer
Verified
482.7k+ views
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.
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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE