What is the Derivative of Ex
Ex here is the natural exponential function which is written as e^x, where x is any real number. To find the derivative of e^x first let us understand what a derivative is. Derivative can be defined as the instantaneous rate of change of a function with respect to one of its variables which in simple words is the slope of the tangent line to the function at a point. The derivative of a function f(x) with respect to ‘x’ is the function f’(x) given by , f’(x) = lim(h?0) [f(x+h) – f(x)]/h where f’(x) is read as ‘f prime of x’. Let us now find what is the derivative of Ex. We need to find the derivative of e^x to get the derivative of Ex. Let us take y= e^x, which is the inverse of y= ln x, for which we can obtain the derivative. y= e^x implies ln y = ln e^x = x. Taking derivatives on both sides of lny = x, we get d[ln y]/dx = d[x]/dx and applying the chain rule to ln y, we get, 1/y. y’ = 1 which gives, y’ = y . So, in this case we can see that the derivative of y is y which means derivative of e^x is itself. We can conclude that the Derivative of e^x is e^x
Let us now find the derivative of e with a functional exponent. Let y = e^u(x), applying the chain rule, we get, d[e^u(x)]/dx= d[e^u(x)]/dx. du(x)/dx which gives us e^u(x). du(x)/dx. Using the derivative of e with a functional exponent, let us find the derivative of E 4x. Here u=4x, d[e^4x]/dx would be, e^u. d[u]/dx. du/dx = d(4x)/dx = 4. So, substituting u=4x and du/dx = 4, we get, derivative of E 4x as 4e^4x.
Derivative of E Ax would be the derivative of e^Ax. Taking g(x)=Ax where A is some constant, let us apply the chain rule to find the derivative of E Ax. As per the chain rule, f(g(x)’ = f’(g(x)).g’(x). Here f(x) = e^Ax and g(x)=Ax. Let us find the derivatives of each functions, f’[g(x)] and g’(x); f’g(x) = e^Ax and g’(x)=A. Substituting these values we get Derivative of E Ax as A.e^Ax
Derivative of E -1 is written as derivative of e^-1. e^-1 is something like e^x where x = -1. We know that the derivative of e^x is e^x and hence the derivative of e^-1 is e^-1. So, Derivative of E -1 is e^-1
