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Calculate the derivative of the function g(x)=e2xg(x) = e^{2x}.
Nov 22, 2023
Calculate the derivative of the function g(x)=e2xg(x) = e^{2x}.
Here is a Step-by-Step solution: #step 1#
To calculate the derivative of the function g(x)=e2xg(x) = e^{2x}, we will use the chain rule. The chain rule is a formula for computing the derivative of the composition of two or more functions.
#step 2#
Let's identify the outer function and the inner function in g(x)g(x). The outer function is the exponential function eue^u where uu is the inner function, which in this case is 2x2x.
#step 3#
Now we apply the chain rule. The derivative of the outer function eue^u with respect to uu is eue^u. Then we multiply this by the derivative of the inner function 2x2x with respect to xx, which is 2.
#step 4#
Putting it all together, the derivative of g(x)g(x) is the derivative of the outer function times the derivative of the inner function: g(x)=e2x2g'(x) = e^{2x} \cdot 2
Answer
The derivative of the function g(x)=e2xg(x) = e^{2x} is g(x)=2e2xg'(x) = 2e^{2x}.
Key Concept
Chain Rule
Explanation
The chain rule was used to differentiate the composite function e2xe^{2x}, resulting in the derivative 2e2x2e^{2x}.
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