Derivative of y= e^x

\(\frac{dy}{dx}\)= e^x

\(\frac{dy}{dx}\)e^u(x))=u’(x) e^u(x)

Optimization Using Exponential Functions

An exponential growth or decay function is a function that grows or shrinks at a constant percent growth rate. The equation can be written in the form

f(x)=a(1+r)^x

Or f(x)= ab^x

• a is the initial or starting value of the function,
• r is the percent growth or decay rate, written as a decimal,
• b is the growth factor or growth multiplier.
• Limit b > 0.

Derivatives of Logarithmic Functions

\(\frac{d}{dx}\) ln(x) =  \(\frac{1}{x}\)

\(\frac{d}{dx}\) ln(u(x) = \(\frac{1}{u(x)}\) u’(x)

\(\frac{d}{dx}\) log_a(x) = \(\frac{1}{x~In~ a}\)

\(\frac{d}{dx}\)   log_a(u(x)) = \(\frac{1}{u(x)~In~a}\) u’(x)

Applications on Logarithmic Functions

Three of the most common applications of exponential and logarithmic functions concern the interest earned on an investment, the  population growth, and carbon dating.