The document discusses the derivative of exponential functions. It provides the general formula that if y = a^x, where a is a constant, then the derivative of y with respect to x is y' = a^x * ln(a). It proves this using properties of logarithms and shows examples of applying the formula to functions like y = 2^x, y = 0.5^x, and y = e^x.

1. DERIVATIVE of EXPONENTIAL

2. GENERAL FORMULA If y = a x , where a = constant or any values then y’ = dy/dx = a x ln a , where ln is log e Similar with y = a f(x) then y’ = dy/dx = a f(x) ln a f’(x)

3. PROVING Let y = a x , then put ln on both sides ln y = ln a x , ln y = x lna Derify both sides & remember derivative of ln x is 1/x 1/y . dy = 1 . lna . dx then rearrange  dy/dx = y lna dy/dx = a x lna

4. Examples Y = 2 x , Y’= 2 x ln2 Y= 0.5 x , Y’= 0.5 x ln0.5 Y = 2 x 2 , Y’= 2 x 2 ln2 (2x) Y= 2 -3x , Y’= 2 -3x ln2 (-3)

5. Examples 2 Y = b x , Y’= b x lnb Y= b x 2 -3x , Y’= b x 2 -3x lnb (2x-3) Y = e x , Y’= e x lne, Since lne = 1, then Y’ = e x For Y = e x , then Y’ = e x

6. Examples 3 Y = e 3x , Y’= e 3x lne (3) Y’= 3e 3x Y = e x 2 -3x , Y’= e x 2 -3x lne (2x-3) Y’=(2x-3)e x 2 -3x