The document provides an overview of derivatives and differentiation in basic calculus, defining derivatives as measures of sensitivity to changes in function inputs. It discusses the derivative of a function with an example and explains the relationship between continuity and differentiability, stating that a differentiable function must be continuous. Additionally, it includes rules for differentiation such as the inverse function rule and reciprocal rule.

1. Derivatives
Basic Calculus 11

2. Derivatives
It is the measure of the sensitivity of the
change of the function value with respect to a
change in its input value.

3. Differentiation
It is the action or process of computing a
derivative of a function.

4. Derivatives
The derivative f(x) with respect to 𝑓′
(x) is defined
as:

5. Example:

6. Example:
f(x) =
𝑥2+1
5𝑥−3
Rewrite the two equations as:
A = 𝑥2
+ 1
B = 5x - 3

7. Example:
By using the quotient rule:
𝑓′
𝑥 =
2𝑥 5𝑥 − 3 − 𝑥2
+ 1 5
(5𝑥 − 3)2
Expand and Group to obtain f’ (x):
𝑓′
𝑥 =
5𝑥2
− 6𝑥 − 5
(5𝑥 − 3)2

8. Continuity and
Differentiability
If y = f(x) is differentiable at a, then f must
also be continuous at a.
Any differentiable function must be
continuous at every point in its domain.

9. Rules in
Differentation

10. Differentiation
It is a process of finding a function
that outputs the rate of change of one
variable with respect to another
variable.

13. Inverse Function Rule

14. Reciprocal Rule
𝟏
𝒗
’ =−
𝒗 𝟐
𝟐