If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Calculus, all content (2017 edition)>Unit 2

Lesson 6: Formal definition of derivative

# Worked example: Derivative as a limit

Discover how to apply the formal and alternate forms of the derivative in real-world scenarios. We'll explore the process of finding the slope of tangent lines using both methods and compare their effectiveness in solving calculus problems. Let's dive into the practical side of derivatives to deepen our understanding. Created by Sal Khan.

## Want to join the conversation?

• i guess im not understanding something here. Its seems in both the formal and alternate limits as h approaches 0 and x approaches e 0/0. So does the limit not exist or did i skip a step?
• The value (e * 0) / 0 doesn't exist, but the limit (ln(e + h) - ln(e)) / h as h approaches 0 does exist. That is why we use limits: they allow us to get a handle on values without breaking algebra.
• Why did someone make the alternate if there was a formal already?
• There are always many ways to solve Math Questions.
Therefore in the subtopic Derivatives there are also many ways to solve the same question.

The Formal way is used when you want a general equation in which you just put a x value and get the Derivative at that particular x.
This can be used when you have to find many Derivatives on the same Function.

But when you want the value at just one particular x value you can use Alternate method.
This will save time for that question.

Whichever you use, you'll get the same answer.

Sal told us the Alternate method so that we can clear our concepts.
Once our concepts are clear we can understand everything nicely. :)
• is this related to the fact that when u zoom into a curve to high levels, the curve appears to be a straight line, and then you can take points on the straight line (they will be very close to each other when zoomed out), and then find the slope?
• Very astute observation! This is exactly how derivatives work. This definition goes into even further (i.e. more rigorous) detail in real analysis.
• What is the derivative of f(x)=ln x?
• What;s the difference between a normal logarithm and a natural logarithm?
• Logarithm has different bases. Usually, we write them with the base. However, for log base of e (a constant similar to pi) is called natural logarithm, and written as ln without a base (and we know it has the base of e). Similarly, there is a case of log (without any base) is understood to be log base of 10. So natural log is just a normal log with a special base (constant e).
• which video tells me to use these formulas to actually find the derivative ?
• Why x approaches 0 in formal definition?
• We can conceptualize the tangent line as being a secant line where the two points get closer an closer. If we put the distances between the secant lines close enough (if the difference between them approaches 0) then we will have a tangent line at exactly one point on the function, which we define to be the derivative.
• how do you find the f'(x) of f(x)=x^e^x ?
this is a question given in an exercise book i have that i am really confused with. so i really hope for someone to explain it to me..
• To make this easier to follow, I will replace f(x) with y, though you don't really need to do this.
y = x^e^x
ln y = ln(x^e^x)
ln y = (e^x)(ln x)
Taking the implicit derivative:
(1/y) dy = [ (e^x)(ln x) + (e^x)(1/x) ] dx
dy/dx = y [ (e^x)(ln x) + (e^x)(1/x) ]
dy/dx = ( x^e^x) [ (e^x)(ln x) + (e^x)(1/x) ]