Practice Natural exponential function
The exponential function eˣ is equal to its own derivative.
It plays a central role in continuous growth and decay models.
Exercises with solutions
The natural exponential function f(x) = e^x has the base e ≈ 2.718. Its special property: it is its own derivative, so f′(x) = e^x. It is positive everywhere and grows faster than any power. In f(x) = a · e^(k·x), k describes the growth rate.
Exercise 1
Differentiate f(x) = e^x.
Show solution and working
f′(x) = e^x
The exponential function is its own derivative, so the expression stays unchanged.
Exercise 2
Differentiate f(x) = e^(2x).
Show solution and working
f′(x) = 2 · e^(2x)
By the chain rule the inner derivative, here 2, is written as a factor in front.
Exercise 3
Can e^x ever become zero?
Show solution and working
No
e^x is always positive. For very negative x the graph approaches the x-axis but never reaches it.
Exercise 4
Calculate f(0) for f(x) = e^x.
Show solution and working
1
Any number to the power 0 gives 1, so e^0 = 1. The graph crosses the y-axis at 1.
Common mistake
Forgetting the chain rule: for e^(2x) the factor 2 must be included, otherwise the derivative is wrong. And believing that e^x can become zero or negative.
Keep learning with MatheJump
Create a free account and explore math practice. Choose from the available exercises to suit your learning needs.