Answer
Verified
479.1k+ views
Hint: The function to be differentiated is a product of two functions of x. Hence, use product rules to differentiate it and then simplify the terms.
Complete step-by-step answer:
We observe that the term to be differentiated is the product of two functions \[{e^{3x}}\] and sin(4x).
We know that to differentiate these terms, we must use the product rule of differentiation.
The product rule of differential calculus states that the differentiation of a product of two functions is the sum of products of one function and the differentiation of the other function and it is given as
follows:
\[(uv)' = uv' + u'v.........(1)\]
where u and v are two functions of x and u’ and v’ are differentiation of u and v with respect to x
respectively.
It is given that,
\[y = {e^{3x}}\sin 4x........(2)\]
We differentiate both sides of the equation (2) to get an expression for \[\dfrac{{dy}}{{dx}}\].
$\Rightarrow$ \[\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {{e^{3x}}\sin 4x} \right)..........(3)\]
Using the formula in equation (1) in equation (3), we get the following:
$\Rightarrow$ \[\dfrac{{dy}}{{dx}} = {e^{3x}}\dfrac{d}{{dx}}\left( {\sin 4x} \right) + \sin 4x\dfrac{d}{{dx}}\left({{e^{3x}}} \right)..........(3)\]
We know that differentiation of sin(ax) is a.cos(ax) and the differentiation of \[{e^{ax}}\] is
\[a{e^{ax}}\]. Using these formulas to simplify equation (3), we get:
$\Rightarrow$ \[\dfrac{{dy}}{{dx}} = {e^{3x}}.4\cos 4x + \sin 4x.3{e^{3x}}\]
Taking \[{e^{3x}}\] as a common term, we get the final expression as:
$\Rightarrow$ \[\dfrac{{dy}}{{dx}} = {e^{3x}}(4\cos 4x + 3\sin 4x)\]
Hence, the answer is \[{e^{3x}}(4\cos 4x + 3\sin 4x)\].
Note: You can easily forget the constant term when differentiating sin(4x) and \[{e^{3x}}\] and the you might get the final answer as \[{e^{3x}}(\cos 4x + \sin 4x)\], which is wrong. This question is an example for application of the product rule of differentiation.
Complete step-by-step answer:
We observe that the term to be differentiated is the product of two functions \[{e^{3x}}\] and sin(4x).
We know that to differentiate these terms, we must use the product rule of differentiation.
The product rule of differential calculus states that the differentiation of a product of two functions is the sum of products of one function and the differentiation of the other function and it is given as
follows:
\[(uv)' = uv' + u'v.........(1)\]
where u and v are two functions of x and u’ and v’ are differentiation of u and v with respect to x
respectively.
It is given that,
\[y = {e^{3x}}\sin 4x........(2)\]
We differentiate both sides of the equation (2) to get an expression for \[\dfrac{{dy}}{{dx}}\].
$\Rightarrow$ \[\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {{e^{3x}}\sin 4x} \right)..........(3)\]
Using the formula in equation (1) in equation (3), we get the following:
$\Rightarrow$ \[\dfrac{{dy}}{{dx}} = {e^{3x}}\dfrac{d}{{dx}}\left( {\sin 4x} \right) + \sin 4x\dfrac{d}{{dx}}\left({{e^{3x}}} \right)..........(3)\]
We know that differentiation of sin(ax) is a.cos(ax) and the differentiation of \[{e^{ax}}\] is
\[a{e^{ax}}\]. Using these formulas to simplify equation (3), we get:
$\Rightarrow$ \[\dfrac{{dy}}{{dx}} = {e^{3x}}.4\cos 4x + \sin 4x.3{e^{3x}}\]
Taking \[{e^{3x}}\] as a common term, we get the final expression as:
$\Rightarrow$ \[\dfrac{{dy}}{{dx}} = {e^{3x}}(4\cos 4x + 3\sin 4x)\]
Hence, the answer is \[{e^{3x}}(4\cos 4x + 3\sin 4x)\].
Note: You can easily forget the constant term when differentiating sin(4x) and \[{e^{3x}}\] and the you might get the final answer as \[{e^{3x}}(\cos 4x + \sin 4x)\], which is wrong. This question is an example for application of the product rule of differentiation.
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
Which are the Top 10 Largest Countries of the World?
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
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Change the following sentences into negative and interrogative class 10 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE