Derivative of e 2x means we have to find the differentiation of e^2x. To understand we should first understand that the
differentiation of e^x is e^x. Now in this case x is replaced by the 2x. So now we will solve the differentiation of e^2x in this way:-
Let y =e^2x, y = e^z
Therefore, z = 2x.
We know that, dy/dx = (dy/dz). (dz/dx)
So dy/dz = e^z
And dz/dx = d (2x)/dx = 2
Hence dy/dx = e^z. 2
By substituting the value of x,
We get dy/dx = (e^2x). 2
This can be easily understood by the following below mentioned examples.
Question 1: Find the differentiation of 2 (e^2x) with respect to x.
Solution: Let y = 2 (e^2x).
We know that d (ky)/dx = k (dy/dx), Here k is constant
So d (2 (e^2x))/ dx = 2 (d (e^2x)/ dx) (Because here 2 is a constant term, hence it comes out from the derivative.
Therefore by definition dy/dx = 2 (e^2x) (2)
So dy/ dx = 4 (e^2x)
Hence, d (2 (e^2x))/ dx = 4 (e^2x).
Question 2: Find the differentiation of x + e^2x with respect to x.
Solution: Let y = x + e^2x
Now first of all we will apply the sum rule of differentiation,
Therefore, dy/dx = dx/dx + d (e^2x)/dx
So dy/dx = 1 + d (e^2x)/dx (Because dx and dx will cancels out)
After solving, dy/dx = 1 + 2 (e^2x)
Hence d(x + e^2x)/ dx = 1 + 2 (e^2x).