§ 4.3. Differentiation of Exponential Functions. Section Outline. Chain Rule for e g ( x ) Working With Differential Equations Solving Differential Equations at Initial Values Functions of the form e kx. Chain Rule for e g ( x ). Chain Rule for e g ( x ). EXAMPLE. Differentiate.
Differentiation of Exponential Functions
This is the given function.
Use the chain rule.
Use the chain rule for exponential functions.
Generally speaking, a differential equation is an equation that contains a derivative.
Determine all solutions of the differential equation
The equation has the form y΄ = ky with k = 1/3. Therefore, any
solution of the equation has the form
where C is a constant.
Determine all functions y = f(x) such that y΄ = 3y and f(0) = ½.
The equation has the form y΄ = ky with k = 3. Therefore,
for some constant C. We also require that f(0) = ½. That is,
So C = ½ and