What is the derivative of e^(2x)?
Understand the Problem
The question is asking for the derivative of the function e^(2x), which is a mathematical concept. To find this, we would apply the chain rule of differentiation.
Answer
The derivative of \( e^{2x} \) is \( 2e^{2x} \).
Answer for screen readers
The derivative of the function ( e^{2x} ) is ( 2e^{2x} ).
Steps to Solve
- Identify the function to differentiate
We have the function ( f(x) = e^{2x} ). We need to find the derivative ( f'(x) ).
- Apply the chain rule
The chain rule states that if you have a composed function ( f(g(x)) ), the derivative is given by ( f'(g(x)) \cdot g'(x) ). In this case, we can identify ( g(x) = 2x ).
- Differentiate the outer function
The derivative of ( e^{u} ) with respect to ( u ) is ( e^{u} ). Here, our outer function is ( e^{g(x)} = e^{2x} ).
- Differentiate the inner function
The inner function ( g(x) = 2x ) has a derivative of ( g'(x) = 2 ).
- Combine the derivatives using the chain rule
Combining both derivatives, we have:
$$ f'(x) = e^{g(x)} \cdot g'(x) = e^{2x} \cdot 2 $$
Thus,
$$ f'(x) = 2e^{2x} $$
The derivative of the function ( e^{2x} ) is ( 2e^{2x} ).
More Information
The function ( e^{2x} ) represents an exponential function, which grows rapidly as ( x ) increases. The derivative, ( 2e^{2x} ), tells us how steep the function is at any point ( x ). Exponential derivatives often retain the same base, multiplied by the constant factor from the exponent.
Tips
- Forgetting to apply the chain rule: Always remember that when you have a composite function, you need to differentiate both the outer and inner functions.
- Confusing the constants: It’s essential to correctly identify and separate the constants during differentiation.
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