Welcome to Mathematics III: Differential Calculus!
Remember differentiation from Mathematics II (differentiating polynomial functions)? In Mathematics III Differential Calculus, your world expands dramatically! You will learn how to differentiate all kinds of functions encountered in senior high school mathematics, including rational functions, radical functions, trigonometric functions, exponential functions, and logarithmic functions. You might think, "There are so many formulas, it looks so hard...", but don't worry! By building up the fundamental rules one step at a time, solving these problems will feel as fun and satisfying as completing a puzzle. Take your time, and let's master them together!
* Note: Basic limit calculations (\(\lim\)) and integration methods are covered in detail in their respective units. Here, we will focus on the theory, calculation techniques, and applications of differentiation.
1. Continuity and Differentiability
Before differentiating a function, let's make sure we clearly understand what it actually means for a function to be "differentiable."
The Relationship Between Continuity and Differentiability
For a function \(y = f(x)\), the derivative (or instantaneous rate of change) at \(x = a\), denoted by \(f'(a)\), is defined as follows:
\(f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\) (or \(\lim_{x \to a} \frac{f(x) - f(a)}{x - a}\))
When this limit exists, we say that the function \(f(x)\) is differentiable at \(x = a\).
Here is an essential theorem to remember:
"If a function \(f(x)\) is differentiable at \(x = a\), then \(f(x)\) is continuous at \(x = a\)."
If a graph is broken or has a gap (discontinuous), it is impossible to draw a tangent line (differentiate). However, the converse is not necessarily true!
[Common Mistake] Assuming "continuous means always differentiable"
Even if a function is continuous, it cannot be differentiated at a sharp corner or "cusp" on the graph.
For example, the graph of \(f(x) = |x|\) is connected at \(x = 0\) (continuous), but the right-hand limit of the difference quotient is \(+1\) and the left-hand limit is \(-1\). Because these one-sided limits do not match, the function is not differentiable at \(x = 0\).
[Key Takeaways]
- Differentiable \(\implies\) Continuous (Always true)
- Continuous \(\implies\) Differentiable (Not always true! Counterexample: \(y = |x|\) at \(x = 0\))
2. Fundamental Differentiation Rules (Product, Quotient, and Chain Rules)
These are powerful rules for computing derivatives in Mathematics III. No matter how complicated an expression looks, you can differentiate it by breaking it down using these rules.
(1) Sum, Difference, and Constant Multiple Rules
- \(\{kf(x)\}' = k f'(x)\) (\(k\) is a constant)
- \(\{f(x) + g(x)\}' = f'(x) + g'(x)\)
- \(\{f(x) - g(x)\}' = f'(x) - g'(x)\)
(2) The Product Rule
\(\{f(x)g(x)\}' = f'(x)g(x) + f(x)g'(x)\)
A great way to remember this rhythm is: "Differentiate the first, leave the second alone, plus leave the first alone, differentiate the second!"
(3) The Quotient Rule
Use the following formulas when differentiating rational functions (where the denominator is non-zero):
- \(\left\{ \frac{1}{g(x)} \right\}' = -\frac{g'(x)}{\{g(x)\}^2}\)
- \(\left\{ \frac{f(x)}{g(x)} \right\}' = \frac{f'(x)g(x) - f(x)g'(x)}{\{g(x)\}^2}\)
Pay close attention to the order of subtraction in the numerator: "(derivative of numerator \(\times\) denominator) \(-\) (numerator \(\times\) derivative of denominator)".
(4) The Chain Rule (Differentiating Composite Functions)
When \(y = f(u)\) and \(u = g(x)\), the derivative of the composite function \(y = f(g(x))\) is given by:
\(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\), or in prime notation, \(\{f(g(x))\}' = f'(g(x)) \cdot g'(x)\)
Differentiate the "outside" function first, then multiply by the derivative of the "inside" function.
Example: Differentiating \(y = (2x + 1)^3\)
Outer derivative: \(3(2x + 1)^2\)
Inner derivative: \((2x + 1)' = 2\)
Therefore, \(y' = 3(2x + 1)^2 \cdot 2 = 6(2x + 1)^2\)
[Common Mistake] Forgetting to multiply by the derivative of the inside!
A frequent error is leaving \(\{(3x - 1)^4\}'\) as simply \(4(3x - 1)^3\). Always remember to multiply by the inner derivative, \(3\), to get \(12(3x - 1)^3\).
3. Derivatives of Various Functions
Here are the differentiation formulas for the new functions introduced in Mathematics III. All of these can be derived directly from the limit definition: \(\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}\).
(1) Derivatives of Trigonometric Functions
- \((\sin x)' = \cos x\)
- \((\cos x)' = -\sin x\) (Don't forget the minus sign!)
- \((\tan x)' = \frac{1}{\cos^2 x}\) (Can be derived using the quotient rule)
(2) Natural Logarithms and Derivatives of Exponential and Logarithmic Functions
A logarithm with base \(e\) (Euler's number, approximately \(2.71828\dots\)), which is defined by the limit \(\lim_{h \to 0} (1+h)^{\frac{1}{h}} = e\), is called the natural logarithm. It is written simply as \(\log x\) (or \(\ln x\)), omitting the base \(e\).
- \((e^x)' = e^x\) (A truly remarkable and elegant function whose derivative is identical to itself!)
- \((a^x)' = a^x \log a\) (\(a > 0, a \neq 1\))
- \((\log x)' = \frac{1}{x}\)
- \((\log_a x)' = \frac{1}{x \log a}\) (\(a > 0, a \neq 1\))
A crucial formula that frequently appears when combined with the chain rule:
- \(\{\log |f(x)|\}' = \frac{f'(x)}{f(x)}\)
[Fun Fact] Why do we use the base of the natural logarithm, \(e\)?
When you differentiate an exponential function with an arbitrary base \(a\), \(a^x\), an extra constant factor of \(\log a\) tags along. However, if you choose the base to be \(e\), then \(\log e = 1\), giving the simplest and neatest derivative possible: \((e^x)' = e^x\). Because it arises so naturally and conveniently when analyzing real-world phenomena, it is called the "natural" logarithm.
4. Equations of Tangent Lines
The slope of the tangent line to the curve \(y = f(x)\) at the point \((a, f(a))\) is the derivative \(f'(a)\). You can apply this directly to the point-slope form of a line learned in Mathematics II:
Equation of the Tangent Line: \(y - f(a) = f'(a)(x - a)\)
Three common scenarios for finding tangent lines:
1. When the point of tangency \((a, f(a))\) is given: Simply substitute the values directly into the formula.
2. When the slope \(m\) is given: Solve the equation \(f'(a) = m\) to determine the \(x\)-coordinate \(a\) of the point of tangency.
3. When the tangent line passes through an external point \((x_0, y_0)\): First, set the unknown point of tangency as \((t, f(t))\) to form the general tangent line equation. Then, substitute \((x_0, y_0)\) into the equation to solve for \(t\).
5. The Second Derivative, Concavity, and Points of Inflection
The function obtained by differentiating the first derivative \(f'(x)\) once more is called the second derivative, denoted as \(f''(x)\) or \(\frac{d^2y}{dx^2}\).
Connecting Derivatives to Increase/Decrease and Concavity
- On intervals where \(f'(x) > 0\), the graph is increasing (sloping upwards to the right)
- On intervals where \(f'(x) < 0\), the graph is decreasing (sloping downwards to the right)
- On intervals where \(f''(x) > 0\), the graph is concave up (shaped like a cup or valley; the tangent slope is increasing)
- On intervals where \(f''(x) < 0\), the graph is concave down (shaped like a cap or hill; the tangent slope is decreasing)
What is a Point of Inflection?
A point on a curve where the concavity changes from concave up to concave down, or from concave down to concave up, is called a point of inflection.
[Key Method] How to find points of inflection:
1. Find the values of \(x\) where \(f''(x) = 0\).
2. Check that the sign of \(f''(x)\) actually changes (from positive to negative, or from negative to positive) across that value of \(x\).
A point \((a, f(a))\) that satisfies both conditions is a point of inflection.
[Common Mistake] Assuming \(f''(a) = 0\) automatically means an inflection point!
Just as \(f'(a) = 0\) does not guarantee a local extremum, \(f''(a) = 0\) does not make it an inflection point if the sign does not change. (For instance, for \(f(x) = x^4\), we have \(f''(0) = 0\), but \(f''(x) \ge 0\) everywhere, meaning the graph remains concave up and does not have an inflection point at \(x = 0\)).
6. Sketching Curve Shapes: Increase, Decrease, and Extrema
When sketching graphs in Mathematics III, systematically construct a sign chart (table of increases and decreases) using these steps:
Step 1: Determine the domain
Always check domain restrictions first: denominators \(\neq 0\), terms inside square roots \(\ge 0\), arguments of logarithms \(> 0\), etc.
Step 2: Calculate \(f'(x)\) and \(f''(x)\)
Find the values of \(x\) where \(f'(x) = 0\) and \(f''(x) = 0\).
Step 3: Construct the sign and concavity chart
Include rows for \(x\), \(f'(x)\), \(f''(x)\), and \(f(x)\). Represent the curve's profile using curved arrows:
- \(f' > 0, f'' > 0\) \(\implies\) Concave up and increasing (rising valley shape: \(\nearrow\) curving downward)
- \(f' > 0, f'' < 0\) \(\implies\) Concave down and increasing (rising crest shape: \(\nearrow\) curving upward)
- \(f' < 0, f'' > 0\) \(\implies\) Concave up and decreasing (falling valley shape: \(\searrow\) curving downward)
- \(f' < 0, f'' < 0\) \(\implies\) Concave down and decreasing (falling crest shape: \(\searrow\) curving upward)
Step 4: Check end behavior, limits at infinity, and sketch the graph
Investigate asymptotes (behavior as \(x \to \pm\infty\) or near the boundaries of the domain) and connect key points with a smooth curve.
7. Applications of Differentiation: Velocity and Acceleration in Linear and Planar Motion
Differentiation is an invaluable tool for analyzing physical motion. Differentiating position with respect to time \(t\) gives velocity, and differentiating velocity with respect to time \(t\) gives acceleration.
(1) Motion Along a Straight Line
Let the position of a point \(\mathrm{P}\) moving along a coordinate line at time \(t\) be \(x = f(t)\):
- Velocity \(v\): \(v = \frac{dx}{dt} = f'(t)\)
- Speed: The magnitude of velocity, \(|v|\)
- Acceleration \(\alpha\): \(\alpha = \frac{dv}{dt} = \frac{d^2x}{dt^2} = f''(t)\)
(2) Motion in a Plane
When the coordinates of a point \(\mathrm{P}\) moving in a coordinate plane are defined parametrically at time \(t\) by \((x, y) = (f(t), g(t))\):
- Velocity vector \(\vec{v}\): In component form, \(\vec{v} = \left(\frac{dx}{dt}, \frac{dy}{dt}\right)\)
- Speed (Magnitude of velocity): \(|\vec{v}| = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\)
- Acceleration vector \(\vec{\alpha}\): In component form, \(\vec{\alpha} = \left(\frac{d^2x}{dt^2}, \frac{d^2y}{dt^2}\right)\)
- Magnitude of acceleration: \(|\vec{\alpha}| = \sqrt{\left(\frac{d^2x}{dt^2}\right)^2 + \left(\frac{d^2y}{dt^2}\right)^2}\)
[Key Distinction] The difference between "Velocity" and "Speed"
While these terms are often used interchangeably in everyday conversation, they have precise meanings in mathematics and physics.
Velocity is a vector quantity that includes direction, meaning it can take negative values. Speed is a scalar quantity representing magnitude, so it is always greater than or equal to \(0\).
Chapter Summary
- Differentiability: Continuity does not guarantee differentiability (watch out for sharp corners/cusps).
- Differentiation Rules: Practice the product, quotient, and chain rules repeatedly until they become second nature.
- Transcendental Functions: Memorize the differentiation formulas for trigonometric, exponential, and logarithmic functions accurately.
- The Second Derivative \(f''(x)\): An indispensable tool for analyzing concavity and locating points of inflection.
- Velocity and Acceleration: Position \(\xrightarrow{\text{differentiate}}\) Velocity \(\xrightarrow{\text{differentiate}}\) Acceleration. Remember: speed is the magnitude (absolute value)!
At first, the calculations might feel extensive and intimidating, but differential calculus is a topic where steady practice yields dramatic improvements. Master the fundamental formulas, and tackle each problem with confidence!