Advanced Differentiation: Specialist Mathematics Unit 3

In Specialist Mathematics, differentiation extends beyond Mathematical Methods to include more complex functions, such as inverse circular functions, and more rigorous applications of the product, quotient, and chain rules.

1. Fundamental Rules of Differentiation

To differentiate advanced functions, proficiency in the three primary rules is essential.

The Chain Rule

Used for composite functions of the form $y = f(g(x))$.
$$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$$
Inverse Function Property: A specific application of the chain rule used when $x$ is defined in terms of $y$:
$$\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}$$

The Product Rule

Used when a function is the product of two differentiable functions: $y = u(x)v(x)$.
$$\frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}$$

The Quotient Rule

Used when a function is the ratio of two differentiable functions: $y = \frac{u(x)}{v(x)}$.
$$\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$$

COMMON MISTAKE: In the quotient rule, the order of terms in the numerator matters because of the subtraction. Always start with $v \frac{du}{dx}$. A common error is swapping them, which results in the negative of the correct derivative.

2. Derivatives of Standard Functions

The following table summarizes the derivatives required for Specialist Mathematics.

VCAA FOCUS: Students are frequently tested on the derivatives of inverse circular functions. Pay close attention to the value of $a$ and the use of the chain rule if the argument is more complex than $\frac{x}{a}$ (e.g., $\sin^{-1}(3x-1)$).

3. Differentiating Rational Functions

A rational function is defined as $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials.

Techniques for Differentiation

1. Direct Quotient Rule: Apply $\frac{v u’ - u v’}{v^2}$ directly.
2. Polynomial Division: If the degree of $P(x) \ge$ degree of $Q(x)$, use long division to rewrite the function first. This often makes differentiation and finding asymptotes easier.
   • Example: $f(x) = \frac{x^2+1}{x} = x + \frac{1}{x}$
   • $f’(x) = 1 - \frac{1}{x^2}$

Applications to Graphing

• Vertical Asymptotes: Occur where $Q(x) = 0$ (and $P(x) \neq 0$).
• Stationary Points: Solve $f’(x) = 0$.
• Nature of Points: Use the first or second derivative test to determine if points are local maxima, minima, or points of inflection.

KEY TAKEAWAY: Rewriting a rational function using division can reveal “oblique” or “non-vertical” asymptotes. For $f(x) = ax + b + \frac{r}{Q(x)}$, the line $y = ax + b$ is the asymptote as $x \to \pm\infty$.

4. Logarithmic and Exponential Variations

Specialist Mathematics requires differentiating logs with absolute values and complex exponents.

The Derivative of $\log_e|f(x)|$

Using the chain rule:
$$\frac{d}{dx}(\log_e|f(x)|) = \frac{f’(x)}{f(x)}$$
The modulus sign $|x|$ ensures the domain is $x \in \mathbb{R} \setminus {0}$, but the derivative formula remains $\frac{1}{x}$.

Complex Exponentials

For functions like $y = e^{g(x)}$:
$$\frac{dy}{dx} = g’(x)e^{g(x)}$$

EXAM TIP: When differentiating functions like $y = \log_e\left(\frac{ex}{ex+1}\right)$, use log laws to expand the expression before differentiating: $\log_e(ex) - \log_e(ex+1)$. This significantly simplifies the calculus.

5. Higher Order Derivatives

The second derivative is the derivative of the first derivative.

• Notation: $f’‘(x)$ or $\frac{d^2y}{dx^2}$.
• Concavity:
  • If $f’‘(x) > 0$, the graph is concave up (holds water).
  • If $f’‘(x) < 0$, the graph is concave down (sheds water).
• Points of Inflection: Occur where concavity changes. A necessary (but not sufficient) condition is $f’‘(x) = 0$. You must check that the sign of $f’‘(x)$ actually changes across the point.

Successive Differentiation Table

STUDY HINT: To find the $n$-th derivative of a function like $f(x) = xe^x$, calculate the first three derivatives and look for a pattern. You can often prove these patterns using Mathematical Induction, which is a common cross-topic exam question.