Welcome to Calculus: Finding the Slope of a Curve

In previous chapters, you learned that the derivative of a function tells us the rate of change or the gradient (slope) of a curve at any given point. While the concept of a limit is the foundation of this (see Section 5.1), we don't always want to do long calculations to find a gradient. In this chapter, we learn the "short-cuts" for differentiating polynomials and how to use those derivatives to find the equations of tangents and normals.

1. The Power Rule: Differentiating Polynomials

The Power Rule is your best friend in calculus. It allows you to find the derivative of any term where \(x\) is raised to a power. According to the official syllabus, for any term \(ax^n\), the derivative is found by multiplying the coefficient by the power and then reducing the power by one.

The Rule: If \(f(x) = ax^n\), then \(f'(x) = anx^{n-1}\).

How to do it (Step-by-Step):

1. Bring the power down: Multiply the number in front (the coefficient) by the current exponent (\(n\)).
2. Reduce the power: Subtract 1 from the original exponent.

Example: Differentiate \(f(x) = 5x^3\).
Multiply \(5\) by the power \(3\) to get \(15\).
Subtract \(1\) from the power \(3\) to get \(2\).
The result is \(f'(x) = 15x^2\).

Important Special Cases:

  • Constants: The derivative of a constant (a number with no \(x\)) is always \(0\). For example, if \(f(x) = 10\), then \(f'(x) = 0\). (Think about it: a horizontal line has a slope of zero!)
  • Linear terms: The derivative of \(ax\) is just \(a\). For example, if \(f(x) = 7x\), then \(f'(x) = 7\).
  • Negative exponents: The rule works for negative integers too! If \(f(x) = x^{-2}\), then \(f'(x) = -2x^{-3}\).

Quick Tip: If you see a polynomial with multiple terms, like \(f(x) = 4x^3 + 2x^2 - 5x + 3\), just differentiate each part separately! The answer would be \(f'(x) = 12x^2 + 4x - 5\).

Key Takeaway:

To differentiate \(ax^n\), multiply by the power and then subtract one from the power. Constants disappear!

2. Tangents: The "Touching" Line

A tangent is a straight line that just touches a curve at a specific point. At that exact point of contact, the tangent and the curve have the exact same gradient.

Finding the Equation of a Tangent:

To find the equation of a tangent at a point \((x_1, y_1)\), you need two things: a point and a gradient (\(m\)).

Step 1: Find the y-coordinate. If the question only gives you \(x\), plug it back into the original function \(f(x)\) to find \(y\).
Step 2: Find the gradient (\(m\)). Differentiate the function to get \(f'(x)\), then plug your \(x\)-value into this derivative.
Step 3: Use the line formula. Use the formula \(y - y_1 = m(x - x_1)\) or \(y = mx + c\) to find the final equation.

Example: Find the tangent to \(f(x) = x^2\) at \(x = 3\).
1. Point: When \(x = 3\), \(y = 3^2 = 9\). So the point is \((3, 9)\).
2. Gradient: \(f'(x) = 2x\). At \(x = 3\), \(m = 2(3) = 6\).
3. Equation: \(y - 9 = 6(x - 3) \implies y - 9 = 6x - 18 \implies y = 6x - 9\).

Did you know? Your Graphic Display Calculator (GDC) can actually draw tangents for you! Look for the "Sketch" or "Tangent" menu while viewing a graph to verify your manual work.

3. Normals: The Perpendicular Line

A normal is a straight line that is perpendicular (at a 90-degree angle) to the tangent at the point of contact.

From your study of functions (Topic 2.1), you should remember that if two lines are perpendicular, their gradients \(m_1\) and \(m_2\) follow this rule:
\(m_1 \times m_2 = -1\)

In simpler terms, the gradient of the normal is the negative reciprocal of the gradient of the tangent.

How to find the Normal's Gradient:

1. Find the tangent gradient (\(m_{tangent}\)).
2. "Flip and switch": Turn the fraction upside down and change the sign.
\(m_{normal} = -\frac{1}{m_{tangent}}\)

Example: If the gradient of the tangent is \(4\), the gradient of the normal is \(-\frac{1}{4}\). If the tangent gradient is \(-\frac{2}{3}\), the normal gradient is \(\frac{3}{2}\).

Common Mistake to Avoid:

Don't forget to find the \(y\)-coordinate! Students often plug the \(x\)-value into the derivative but forget that they still need the original \(y\)-value from the original function to write the equation of the line.

4. Summary of Steps for Tangents and Normals

Whether you are finding a tangent or a normal, the workflow is very similar. Use this checklist for exam questions:

  1. Differentiate: Find \(f'(x)\) using the Power Rule.
  2. Find the Gradient: Plug the given \(x\)-value into \(f'(x)\).
    • For a Tangent: Use this value as your \(m\).
    • For a Normal: Use \(-\frac{1}{m}\).
  3. Find the Point: If you don't have the \(y\)-coordinate, plug \(x\) into the original function \(f(x)\).
  4. Write the Equation: Use \(y - y_1 = m(x - x_1)\).

Quick Review:
- Derivative of \(x^n\) is \(nx^{n-1}\).
- Tangent gradient = \(f'(x)\).
- Normal gradient = \(-\frac{1}{f'(x)}\).
- Always ensure your final answer is in the form requested (usually \(y = mx + c\) or \(ax + by + d = 0\)).

Note for HL Students: While SL focuses on integer exponents, you will apply these same tangent and normal principles to more complex functions like \(\sin x\), \(e^x\), and \(\ln x\) in later chapters (see Section 5.9).