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Question: Find the derivative of \(f\left( x \right) = {e^{2x}}\)?...

Find the derivative of f(x)=e2xf\left( x \right) = {e^{2x}}?

Explanation

Solution

For the above question we have been given a composite function to find its derivative and we know that to find the derivative of a composite function we will have to use the chain rule of derivative which states that the derivative of a composite function, ddx(f(g(x)))=f′(g(x))g′(x)\dfrac{d}{{dx}}\left( {f\left( {g\left( x \right)} \right)} \right) = f'\left( {g\left( x \right)} \right)g'\left( x \right).

Complete step by step answer:
We have been asked to find the derivative of the function, f(x)=e2xf\left( x \right) = {e^{2x}}.
Differentiation is used to find rates of change. For example, Differentiation allows us to find the rate of change of velocity with respect to time (which gives us acceleration). The concept of differentiation also allows us to find the rate of change of the variable x with respect to variable y, which is plotted on a graph of y against x, which is known to be the gradient of the curve.

Since it is a composite function, so we will use the chain rule of derivative to find its derivatives and the chain rule states that the derivative of a function f(g(x))f\left( {g\left( x \right)} \right) is equal to f′(g(x))g′(x)f'\left( {g\left( x \right)} \right)g'\left( x \right).
First, we will need to use the Chain Rule
dydx=dydu×dudx\dfrac{{dy}}{{dx}} = \dfrac{{dy}}{{du}} \times \dfrac{{du}}{{dx}}
So, the differentiation of f(x)=e2xf\left( x \right) = {e^{2x}} is shown as follows by applying the chain rule,
⇒f′(x)=ddx(e2x)\Rightarrow f'\left( x \right) = \dfrac{d}{{dx}}\left( {{e^{2x}}} \right)
Since we know the derivative of ex{e^x} is ex{e^x}. Then,
⇒f′(x)=e2x×ddx(2x)\Rightarrow f'\left( x \right) = {e^{2x}} \times \dfrac{d}{{dx}}\left( {2x} \right)
Now, the derivative of x is 1.
⇒f′(x)=2e2x\Rightarrow f'\left( x \right) = 2{e^{2x}}
Hence, the required derivative of the given function is equal to 2e2x2{e^{2x}}.

Note: Before applying the chain rule, you have to take care that the question involves function inside the given function, only then the chain rule can be applied. If the question involves a chain rule and we are comfortable using it, then we can directly differentiate each function separately and write the answer.