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Can the derivative of an exponential function be calculated with logs base something other than e? Like base 10 or 2?
theorem4.5.9 said:You can, but you will get an extra constant from the chain rule. Just use a change of base formula for log or rewrite an exponential in some other base and the term comes right out.
##e## is important precisely because it's the only base of the exponential where after differentiating the constant is ##1##.
The derivative of an exponential function with a natural logarithm is equal to the original function multiplied by the derivative of the exponent. In other words, if the function is f(x) = e^(lnx), its derivative would be f'(x) = x * (1/x) = 1.
To find the derivative of an exponential function with a base other than e, you can use the logarithmic differentiation method. This involves taking the natural logarithm of both sides of the function, using logarithm rules to simplify, and then finding the derivative using the chain rule.
The derivative of a logarithm with a base other than e is equal to the natural logarithm of the base multiplied by the original function's derivative. For example, if the function is f(x) = log2x, its derivative would be f'(x) = (ln2) * (1/x) = ln2/x.
Yes, the power rule can be used to find the derivative of an exponential function with a natural logarithm. The power rule states that the derivative of x^n is equal to n * x^(n-1). In the case of an exponential function with a natural logarithm, the derivative would be e^(lnx) * (1/x) = x * (1/x) = 1.
To find the derivative of a function with multiple exponential and logarithmic terms, you can use the product rule and chain rule in combination. First, use the product rule to find the derivative of each individual term, and then use the chain rule to take the derivative of the exponent or logarithm within each term.