# Calculus 1: Exponential and Logarithmic Functions

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 logarithmic differentiation to find the derivative in the following example

$g(x) = \log_{3}(2x^{2} - 5x)$

Find the derivative of $h(x) = 8^{x}\log_{9}x$

Use the properties of logarithms to show that the derivative of $\log_{a}x = \frac{1}{(\ln{a})x}$

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.

Differentiate $f(x) = \ln{6x^2}$

Find the derivative of $f(x) = \frac{\ln{(2x)}}{x^4}$