Use the definition of \(e\) as the unique positive number for which \(\lim_{h \rightarrow 0}\frac{e^{h} - 1}{h} = 1\) and the definition of the derivative to show that derivative of the exponential function, \(f(x) = e^x\) is equal to \(e^x\)
Determine the derivative of \(f(x) = 2x^5 - 3e^{6x}\)
Find the derivative of \(f(x) = x^{3}e^{-2x}\)
Determine the slope of the tangent line to the function \(f(x) = 2e^{-3x}\)at \((0,2)\)
For the following problem, find the derivative of \(f(x) = 5^{x^{3} - 4}\)
Use implicit differentiation to show that the derivative of \(\ln{x} = \frac{1}{x}\) for \(x > 0\) Note that many classes introduce logarithmic differentiation before implicit differentiation.
Use the properties of logarithms to show that the derivative of \(\log_{a}x = \frac{1}{(\ln{a})x}\)
Differentiate \(f(x) = \ln{6x^2}\)
Use logarithmic differentiation to find the derivative in the following example\(g(x) = \log_{3}(2x^{2} - 5x)\)
Find the derivative of \(f(x) = \frac{\ln{(2x)}}{x^4}\)
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