BackDifferentiation Rules and Applications
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Section 3.3: Differentiation Rules
Introduction
This section covers the fundamental rules for differentiating functions, including the power, constant multiple, sum, product, and quotient rules. These rules are essential for finding derivatives of various types of functions encountered in calculus.
Powers, Multiples, Sums, and Differences
Derivative of a Constant Function: The derivative of any constant function is zero.
Power Rule: For any real number n, the derivative of is:
Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function.
Sum and Difference Rules: The derivative of a sum or difference is the sum or difference of the derivatives.
Examples
Example 1a:
Example 1b:
Example 1c:
Example 1d:
Example 1e:
Example 1f: Simplify numerator:
Derivative of Exponential Functions
Exponential Rule: The derivative of is $e^x$.
Constant Multiple:
Examples
Example 2a:
Example 2b:
Equations of Tangent Lines
The equation of the tangent line to a curve at the point is:
Example 3a: , at Tangent line:
Example 3b: , at Tangent line:
Horizontal Tangents
A tangent line is horizontal where .
Example 4: Set : Points: and
Product Rule
The derivative of a product is not the product of the derivatives. The correct rule is:
Or, in function notation:
Examples
Example 5a: Let , ,
Example 5b:
Quotient Rule
The derivative of a quotient is not the quotient of the derivatives. The correct rule is:
Or, in function notation:
Examples
Example 6a: , ,
Example 6b: , ,
Example 6c: ,
Equations of Tangent Lines (Product/Quotient Functions)
Example 7a: at , , At : , , $u' = 5$, Tangent line:
Example 7b: at From previous, At : , Tangent line:
Higher Derivatives
If is differentiable, then its derivative may also be differentiable. The derivative of $f'(x)$ is called the second derivative, denoted or .
Notation:
Examples
Example 8:
Example 9: (Graphical identification of , , , ) Additional info: When given graphs of a function and its derivatives, identify the original function by looking for the curve with the most inflection points, and match derivatives by their increasing order of oscillation and zero crossings.
