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Calculus 1: Exponential and Logarithmic Functions

Use the definition of ee as the unique positive number for which limh0eh1h=1\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)=exf(x) = e^x is equal to exe^x

Determine the derivative of f(x)=2x53e6xf(x) = 2x^5 - 3e^{6x}

Find the derivative of f(x)=x3e2xf(x) = x^{3}e^{-2x}

Determine the slope of the tangent line to the function f(x)=2e3xf(x) = 2e^{-3x}at (0,2)(0,2)

For the following problem, find the derivative of f(x)=5x34f(x) = 5^{x^{3} - 4}

Use logarithmic differentiation to find the derivative in the following example

g(x)=log3(2x25x)g(x) = \log_{3}(2x^{2} - 5x)

Find the derivative of h(x)=8xlog9xh(x) = 8^{x}\log_{9}x

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

Use implicit differentiation to show that the derivative of lnx=1x\ln{x} = \frac{1}{x} for x>0x > 0

Note that many classes introduce logarithmic differentiation before implicit differentiation.

Differentiate f(x)=ln6x2f(x) = \ln{6x^2}

Find the derivative of f(x)=ln(2x)x4f(x) = \frac{\ln{(2x)}}{x^4}