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### Course: AP®︎/College Calculus AB>Unit 2

Lesson 6: Derivative rules: constant, sum, difference, and constant multiple: introduction

# Basic derivative rules: find the error

We analyze two students' attempts to differentiate linear functions using basic rules. One student forgets a negative sign, while another incorrectly assumes the derivative of a product equals the product of derivatives. By identifying common mistakes, we help improve our understanding of differentiation.

## Want to join the conversation?

• why the derivative the same as the differentiation.
• Differentiation is a process, it is what you do to calculate a derivative.
The derivative is a function, it is the result of differentiation.
EG
f(x)=x² - - - this is the original function
d/dx[f(x)=x²] - - - this is the original function inside the differentiation operator
f'(x)=2x - - - this function is the derivative of the original function. We calculated it using the process of differentiation.
So . . .
d/dx[f(x)=x²] - - - differentiation
f'(x)=2x - - - derivative
• For a linear equation in the form y=mx+b, the coefficient of the x term is the slope of the line. Since the derivative of a function is the slope at any given point and since the slope of a line is always constant, is the derivative not always equal to the coefficient of the x term? Why do we take all of these additional steps?
• Is to proof the basic differentiantion rules. Lineal functions are intuitive, but when you are dealing with more complex functions like x^3+x^2+e^x-log_2(x), they become really usefull.
• How does d(x)/dx turn out to be 1?
• It's the rate of change, or slope, of the function f(x)=x, which is a straight line of slope 1.
• I thought derivatives were for particular points of a function, not whole functions themselves. Why didn't he choose a certain x-value of each function to differentiate?
• That's the cool thing about lines! The derivative of any linear function is a constant, meaning no matter what 𝑥-value you choose, the derivative is always the same. For instance, the derivative of 𝑓(𝑥) = 5𝑥 is 𝑓'(𝑥) = 5. This is 5 no matter what 𝑥 is! Informally, we say that the slope of a line is constant everywhere. Comment if you have questions!
• Why Hannah did is wrong?
• Let 𝑓(𝑥) = 8 and 𝑔(𝑥) = 𝑥
⇒𝑓 '(𝑥) = 0 and 𝑔'(𝑥) = 1

Then 𝑑∕𝑑𝑥[8𝑥] = 𝑑∕𝑑𝑥[𝑓(𝑥)⋅𝑔(𝑥)]

Applying the product rule, we get
𝑓 '(𝑥)⋅𝑔(𝑥) + 𝑓(𝑥)⋅𝑔'(𝑥)

This is not equivalent to
𝑓 '(𝑥)⋅𝑔'(𝑥)
which is what Hannah did.
• at , derivative of 5*f(x) for any x, x is an actual point which means is a constant right?
then why sal says is 1? shouldn't derivative of any constant going to be 0?
i understand graphicly slop of y=x is going to be 1 but not algebraicly
• To find a derivative at a point, we don't plug in the point, solve, then take the derivative and get 0. In doing so, we assume that the function is equal to that constant value everywhere, which is untrue (in general).

The derivative of the function y=x at the point a is the limit as a→x of (a-x)/(a-x), which is the limit of the constant 1, which is 1.
• ``Why aren't we writing dy/dx but d/dx?``