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Preliminary Derivatives

CSCA Preliminary Derivatives study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.

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Preliminary Derivatives

1. Core Concepts: Definition of Derivative

The **Derivative** describes the rate at which a function changes at a certain point, known as the **Instantaneous Rate of Change**.

* **Algebraic Definition**: For a function $y = f(x)$, as $x$ changes by $\Delta x$, $y$ changes by $\Delta y$. The derivative is the limit:

$$ f'(x_0) = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} $$

* **Geometric Meaning**: $f'(x_0)$ represents the **slope of the tangent line** to the curve $y=f(x)$ at the point $(x_0, f(x_0))$.

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2. Basic Differentiation Rules

For the CSCA exam, memorizing these rules is more practical than using the limit definition every time:

1. **Constant Rule**: $(C)' = 0$

2. **Power Rule**: $(x^n)' = n \cdot x^{n-1}$ (Most Important!)

3. **Sum/Difference Rule**: $[f(x) \pm g(x)]' = f'(x) \pm g'(x)$

4. **Constant Multiple Rule**: $[cf(x)]' = c \cdot f'(x)$

* **Common Derivatives**:

* $(\sin x)' = \cos x$

* $(\cos x)' = -\sin x$

* $(e^x)' = e^x$

* $(\ln x)' = \frac{1}{x}$

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3. Applications of Derivatives

Derivatives are powerful tools for analyzing the behavior of function graphs:

1. **Tangent Line Equation**:

The slope at $(x_0, y_0)$ is $k = f'(x_0)$. The equation is:

$$ y - y_0 = f'(x_0)(x - x_0) $$

2. **Monotonicity**:

* If $f'(x) > 0$, $f(x)$ is **Monotonically Increasing**.

* If $f'(x) < 0$, $f(x)$ is **Monotonically Decreasing**.

3. **Extrema (Max/Min)**:

* **Stationary Point**: Solve $f'(x) = 0$ to find candidate points.

* **Test**: If $f'(x)$ changes from positive to negative, it's a **Maximum**; if negative to positive, it's a **Minimum**.

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4. Typical Examples

**Example 1 (Tangent Line)**: Find the tangent line equation for $y = x^2$ at $(1, 1)$.

**Solution**:

1. Differentiate: $y' = (x^2)' = 2x$.

2. Find Slope: At $x=1$, $k = y'|_{x=1} = 2 \times 1 = 2$.

3. Equation: Using point-slope form, $y - 1 = 2(x - 1) \Rightarrow y = 2x - 1$.

**Example 2 (Monotonicity & Extrema)**: Analyze $f(x) = x^3 - 3x$.

**Solution**:

1. Differentiate: $f'(x) = 3x^2 - 3 = 3(x^2 - 1)$.

2. Set $f'(x) = 0$: $x = 1$ or $x = -1$.

3. Analyze Signs:

* $x < -1$: $f'(x) > 0$ (Increasing)

* $-1 < x < 1$: $f'(x) < 0$ (Decreasing)

* $x > 1$: $f'(x) > 0$ (Increasing)

4. Conclusion:

* Increasing on $(-\infty, -1)$ and $(1, +\infty)$. Decreasing on $(-1, 1)$.

* Local Maximum at $x=-1$, Local Minimum at $x=1$.

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5. Common Pitfalls

* **Formula Errors**: Forgetting the coefficient in the Power Rule (e.g., thinking $(x^3)' = x^2$ instead of $3x^2$).

* **Tangent vs. Normal**: Confusing the Tangent line with the Normal line (slope $-1/k$).

* **Stationary Points**: Not all points where $f'(x)=0$ are extrema (e.g., $y=x^3$ at $x=0$). Always check sign changes.