Derivative of e 2x

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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).

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