DERIVATIVE OF e TO THE POWER x

Question :

Find the derivative of e^x with respect to x.

or

Given y = e^x, find dy/dx.

(x and y are variables and e is a constant)

Solution :

In y = e^x, we have constant e in base and variable x in exponent.

y = e^x

Take logarithm on both sides.

lny = lne^x

Apply the power rule of logarithm on the right side.

lny = xlne

(The base of a natural logarithm is e, lne is a natural logarithm and its base is e)

lny = xln_ee

In a logarithm, if the base and argument are same, its value is 1. In ln_ee, the base and argument are same, so its value is 1.

lny = x(1)

lny = x

Find the derivative with respect to x.

Therefore, the derivative e^x is e^x.

Example 1 :

Find the derivative e^mx with respect to x, where m is a constant.

Solution :

The derivative of e^mx is e^mx, further, find the derivative of the exponent mx with respect to x. By chain rule. derivative of mx with respect to x is m(1) = m.

So, the derivative e^mx with respect to x is me^x.

Example 2 :

Find the derivative e^-x with respect to x.

Solution :

The derivative of e^-x is e^-x, further, find the derivative of the exponent -x with respect to x. By chain rule. derivative of -x with respect to x is -1.

So, the derivative e^-x with respect to x is -e^-x.

Example 3 :

Find dy/dx :

Solution :

Example 4 :

Find dy/dx :

Solution :

Example 5 :

Find dy/dx :

y = e^sinx

Solution :

Example 6 :

Find dy/dx :

y = e^cosx

Solution :

Example 7 :

Find dy/dx :

y = e^tanx

Solution :

Example 8 :

Find the derivative e² with respect to x.

Solution :

The derivative of e² is e², further, find the derivative of the exponent 2 with respect to x. By chain rule. derivative of 2 with respect to x is 0.

y = e²

dy/dx = e²⋅ (0)

dy/dx = 0

So, the derivative e² with respect to x is zero.

Note :

We know that e is a mathematical constant and number 2 is a constant. So, e² is a constant. Since the derivative of a constant is zero, the derivative e² is zero.

Derivative of e to the power of any constant is zero. 

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