All exponential functions have the form a x, where a is the base. Therefore, to say that the rate of growth is proportional to its size, is to say that the derivative of a x is proportional to a x. You know that ln (e^x) simplifies down to x, and its derivative is 1, but use the chain rule instead. You'll get (1/e^x) * d/dx (e^x) = 1. Solve for d/dx (e^x) by multiplying both sides by e^x, and you'll solve.
In order to differentiate the exponential function. F (x) = a^x, f (x) = ax, we cannot use power rule as we require the exponent to be a fixed number and the base to be a variable. To differentiate an exponential function, we use a fundamental rule from calculus which states: Here, ln(a) represents the natural logarithm of the base a. Let’s break down how this formula. We will learn that the derivative of an exponential function is proportional to the value of the function itself. The constant of proportionality is found using the base of the exponential.
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