Derivative of e^x
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1) The derivative rule for inverses states that if a function f is differentiable on an interval and its derivative f' is never zero on that interval, then the inverse function f^-1 is also differentiable on that interval. The derivative of the inverse is equal to the reciprocal of the derivative of the original function. 2) As an example, the inverse of the natural logarithm function ln(x) is the exponential function e^x. 3) By applying the inverse derivative rule, we can show that the derivative of the exponential function e^x is itself, or de^x/dx = e^x.
Derivative of e^x
- 1. Derivative of Recall The Derivative Rule for Inverses If is differentiable at every point of an interval and df dx is never zero on , then 1 f − is differentiable at every point of the interval . The value of 1 df dx − at any particular point ( )f a is the reciprocal of the value of df dx at a. ∴ = 1 We know that = ⇔ log = Example 2 = 8 ⇔ log 8 = log 2 = 3log 2 = 3
- 2. Let = ln = log# We find the inverse of = ln = log# = ln = log# ⇒ %& = ∵ = ⇔ log = Inter change x and y = % ∴ = % ∴ The inverse of = ln is = %
- 3. Let us find the Derivative of at = ( Let ) = * + = And observe that * ( = ,- ( = ( ,- = ( We know that ∴ . /*0+ / 1 * 2 = + . /* / 1 32 4 / / 5 ( = /* + / ( *6 (7 = + /* / ( = + / / ,- 3 ( = + . + 1 3 ( = + +/ ( = ( ∴ 4 / / 5 ( = ( Here t is arbitrary Derivative of ∴ / / =