Derivative of e^x | Understanding the Chain Rule in Calculus

(d/dx) e^x =

To find the derivative of the function e^x with respect to x, we can use the chain rule from calculus

To find the derivative of the function e^x with respect to x, we can use the chain rule from calculus. The chain rule states that if we have a composite function f(g(x)), then the derivative df/dx is given by df/dx = (df/dg) * (dg/dx).

Applying the chain rule to e^x, we can consider e^x as the composite function f(g) where g = x. In this case, f(u) = e^u and g(x) = x.

First, let’s find the derivative df/du of f(u) = e^u. The derivative of e^u with respect to u is simply e^u itself. Therefore, df/du = e^u.

Next, let’s find the derivative dg/dx of g(x) = x. The derivative of x with respect to x is 1. Therefore, dg/dx = 1.

Finally, we can substitute these derivatives into the chain rule formula to find the derivative of e^x: df/dx = (df/du) * (dg/dx). Plugging in the values, we get (d/dx) e^x = e^u * 1 = e^x.

Therefore, the derivative of e^x with respect to x, denoted as (d/dx) e^x, is simply e^x itself.

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