Lecture from: 22.04.2024 | Video: Video ETHZ
Differentiable Functions
Clicker Question: Finding a Minimum
Consider the function . We are at the point , where .
Observation: A minimum for this function exists because is continuous, and as , (the term dominates). This means the function must “turn around” somewhere.
The graph shows . If we take a small step to the right from , say to , the function value is slightly less than . This means .
- Question: Starting at , should we move left or right to find a minimum of ?
- Answer: The graph suggests that moving to the right might lead us towards a minimum, because the function is decreasing in the immediate vicinity to the right of (since for a positive ). (This is a baby example of an idea called Gradient Descent.)
Follow-up Question: How could we have made this decision (left or right) if we didn’t have the graph? This is what we want to explore!
Let’s look at a broader view of the function in this example:
It seems our initial step to the right from led us to a minimum, but maybe not the overall (global) minimum.
Question: How can we find or describe minima if we don’t have a picture of the function?
Unfortunately, the minimum we found by moving slightly to the right from was only a local minimum, not the true “global” minimum of the function.
Key Question: What conclusions can we draw about the global behavior of a function based on its local behavior?
Goal of This Chapter
- Investigate the local change of functions. To do this, we’ll use local approximation by straight lines (these lines are called “tangents”).
- Direct Applications:
- Determining the monotonicity of functions (where they are increasing or decreasing).
- Finding local extrema (minima and maxima).
- Taylor Approximation (approximating functions with polynomials of higher order).
The Derivative: Definition and Elementary Consequences
Let , , and be an accumulation point of .
We want to define the “slope of the tangent line” to at as the limit of the “slopes of secant lines.”
The slope of the secant line between and is given by the difference quotient: This difference quotient should provide a good approximation for the slope of at if is close to .
Definition: Differentiability
Let , , and be an accumulation point of . The function is differentiable at if the limit exists in (i.e., it’s a finite real number). If this limit exists, it is called the derivative of at and is denoted by .
Remark: Alternative Limit Form for the Derivative
It is often convenient to set . As , we have . Then the derivative can also be expressed as:
Remark: The Tangent Line
If is differentiable at , then the line defined by the equation is called the tangent line to at . This line is the best linear approximation of near .
The “approximation error” converges to faster than as . Specifically, it holds that: Exercise: Verify this limit using the definition of and .
Theorem: Reformulation of Differentiability
Let , and let be a function. Suppose is an accumulation point of .
Then is differentiable at if and only if there exists a function that is continuous at and satisfies the equation:
In this case, the value of at gives the derivative:
Explanation: What is doing?
Here’s an intuitive explanation of what this is doing:
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Rearranging the Formula: For any in the domain , you can rearrange the formula to solve for : This fraction, , is the slope of the secant line connecting the point and any other point on the graph of . So, for , exactly represents this average slope between and .
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Connecting to Differentiability: Satz 4.1.4 states that a function is differentiable at if and only if such a function exists that is defined on and is continuous at .
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The Role of Continuity: The condition that is continuous at means, by Definition 3.2.1 or its characterization in Satz 3.2.4, that the limit of as approaches exists and is equal to . That is, .
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What Must Be: Now, let’s combine these ideas. Since for , and exists and equals , this means: By the definition of the derivative (Definition 4.1.1), the limit on the left is exactly . Therefore, if is differentiable at and satisfies the condition in Satz 4.1.4, it must be that .
Why is this important?
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The formula:
looks like the linear approximation of , but with a twist - the slope depends on , not just a fixed derivative.
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If is continuous at , it ensures that the “slopes” smoothly approach the true derivative. This gives us differentiability.
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Key idea: Differentiability at a point is equivalent to being able to write the function locally as a linear approximation with a continuous slope function .
Sketch of Proof
- Define as above.
- If is differentiable at , then the difference quotient converges to , so . That makes continuous at , and the formula holds.
- Conversely, if the formula holds and is continuous at , then the difference quotient equals , and: So the derivative exists, and equals .
Corollary: Differentiability Implies Continuity
If is differentiable at , then is continuous at .
Proof
If is differentiable at , then by the theorem above, where is continuous at .
Then . Since , is continuous at .
Examples of Derivatives
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Constant Function: (for some constant ) for all . Then . So, for all .
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Identity Function: for all . Then . So, for all .
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Quadratic Function: for all . . So, for all .
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Absolute Value Function: for all . Is differentiable at ? Consider the limit . If , . So, . If , . So, . Since the right-sided limit (1) and the left-sided limit (-1) are not equal, the overall limit does not exist. Therefore, is not differentiable at .
For any other point , is differentiable: If , then for near , , so . If , then for near , , so . So, for , .
Definition: Differentiable Function
Let , . The function is differentiable (on ) if is differentiable at all accumulation points of with .
Remark: Typically, is a union of intervals with endpoints . In this case, every point of is an accumulation point of .
In this case, for a differentiable function , we get a new function , called the derivative of .
(Example: ).
Examples of Differentiable Functions
- is differentiable, and .
- are differentiable. , .
Proof for
Consider for . For ,
Let . This is a power series (in ) with radius of convergence .
Since is a convergent power series, it is continuous at (by Theorem 3.7.11).
Therefore, . So, .
Thus, .
(For , see script.)
Rules for Differentiation
Theorem: Differentiation Rules
Let , be an accumulation point of , and . Let be differentiable at . Then:
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Linearity of the Derivative: The functions and are differentiable at , and
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Product Rule: The function is differentiable at , and
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Quotient Rule: If , then is an accumulation point of , and the function is differentiable at (on this new domain), with
Proof (using Theorem 4.1.4 - Reformulation of Differentiability)
Since and are differentiable at , there exist functions , continuous at , such that for all :
- with
- with
Sum for 1
. Since and are continuous at , their sum is also continuous at .
By Theorem 4.1.4, is differentiable at , and .
Product for 2
The term is continuous at because are continuous at , and as .
Specifically, .
By Theorem 4.1.4, is differentiable at , and .
Properties (1) and (2) are proven. For (3), see script.
Examples: Applying Differentiation Rules
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Power Rule (by induction): For , for all .
- Base Case (n=1): . (From Example 4.1.6)
- Inductive Step: Assume holds for some . Consider . Using the product rule: (by inductive hypothesis) . The formula holds.
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Tangent Function: is differentiable in its domain . Using the quotient rule: . So, . This holds for .
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Cotangent Function: Analogous to (2). . for .
Continue here: 17 Chain Rule, Derivatives of Inverse Functions, Central Theorems (Local Extrema, Rolle’s Theorem, Mean Value Theorem)