the derivative e⁵log x is equal to
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Answer:
Derivative of ex: Proof and Examples
The exponential function is one of the most important functions in calculus. In this page we'll deduce the expression for the derivative of ex and apply it to calculate the derivative of other exponential functions.
Our first contact with number e and the exponential function was on the page about continuous compound interest and number e. In that page, we gave an intuitive definition of number e, and also an intuitive definition of the exponential function.
We also deduced an alternative expression for the exponential function. The new expression for the exponential function was a series, that is, an infinite sum.
You may ask, the limit definition is much more compact and simple than that ugly infinite sum, why bother?
It turn out that the easiest way to deduce a rule for taking the derivative of ex is using that infinite series representation. Why is that? The series expression for ex looks just like a polynomial.
We can generalize the idea of a polynomial by allowing an infinite number of terms, just like in the expression for the exponential function. An infinite polynomial is called a power series.
The neat thing about a power series is that to calculate its derivative you proceed just like you would with a polynomial. That is, you take the derivative term by term. Let's do that with the exponential function.
Step-by-step explanation:
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