6. Step-by-Step Examples

Example 1: Find the derivative of 𝑓(𝑥) = 5𝑥^4 − 3𝑥^2 + 2. Using the power rule and sum rule: 𝑓′(𝑥) = 5⋅4𝑥^3 − 3⋅2𝑥^1 + 0 = 20𝑥^3 − 6𝑥.

In this example, we are tasked with finding the derivative of the function f(x) = 5x^4 - 3x^2 + 2. We apply two differentiation rules: the power rule and the sum rule.

1. Power Rule: This rule states that if you have a term in the form of x^n, the derivative is n * x^(n-1). We apply this rule to each term of the polynomial.
   • For 5x^4, we multiply 5 by the exponent 4 which equals 20, and then reduce the exponent by 1, giving us 20x^3.
   • For -3x^2, we multiply -3 by 2 giving us -6 and reduce the exponent, resulting in -6x.
   • The term +2 is a constant, and its derivative is 0.
2. We then combine these results to get the complete derivative: f′(x) = 20x^3 - 6x.

Think of this process like a factory producing two different products (the terms 5x^4 and -3x^2) with different production rates. The power rule helps us understand how fast each product quantity is changing as we change the number of workers (x). For each adjustment in workforce, we determine the new rate of production (the derivative) for overall efficiency.

Example 2: Find the slope of the tangent line to 𝑦 = 𝑥^3 − 2𝑥 at 𝑥 = 1.

1. Find derivative: 𝑦′ = 3𝑥^2 − 2.
2. Evaluate at 𝑥 = 1: 𝑦′(1) = 3(1)^2−2 = 3−2 = 1. The slope is 1.
3. Find 𝑦 at 𝑥 = 1: 𝑦(1) = 1−2 = −1.
4. Equation of tangent line: 𝑦−(−1) = 1(𝑥−1) ⟹ 𝑦+1 = 𝑥−1 ⟹ 𝑦 = 𝑥 − 2.

In this example, we want to find the slope of the tangent line to the curve defined by y = x^3 - 2x at the specific point where x = 1.

1. Step 1: Differentiate the Function: Start by finding the derivative of the function y = x^3 - 2x, which tells us how steep the curve is at any point. Using the power rule:
   • The derivative is y' = 3x^2 - 2.
2. Step 2: Evaluate the Derivative at x = 1: Substitute x = 1 into the derivative to find the slope of the tangent line at that point:
   • y'(1) = 3(1)^2 - 2 = 3 - 2 = 1. The slope of the tangent line is 1.
3. Step 3: Find the y-value at x = 1: We need the point on the curve where x = 1. Substitute x = 1 into the original function:
   • y(1) = 1 - 2 = -1. So the point is (1, -1).
4. Step 4: Write the Equation of the Tangent Line: Now that we have the slope (1) and a point (1, -1), we can write the equation of the tangent line using the point-slope formula y - y₁ = m(x - x₁):
   • y - (-1) = 1(x - 1) simplifies to y = x - 2.

Imagine you're driving along a curvy road represented by the function y = x^3 - 2x. You want to know how steep the road is at a specific point (x = 1). The slope of the tangent line gives you the current incline of the road right beneath your car, similar to using a spirit level to see the angle of a slope. The steps we took to find that slope reflect figuring out how steep your incline is based on the curvature of the road.

Example 3: Find the derivative using the definition for 𝑓(𝑥) = 𝑥^2. (𝑎 + ℎ)^2 − 𝑎^2 𝑎^2 + 2𝑎ℎ + ℎ^2 - 𝑎^2 𝑓′(𝑎) = lim = lim = lim = lim(2𝑎 + ℎ) = 2𝑎. ℎ→0 ℎ ℎ→0 ℎ ℎ→0 Thus, 𝑓′(𝑥) = 2𝑥.

In this example, we demonstrate how to find the derivative of f(x) = x^2 using its definition based on limits.

1. Understanding the Limit Definition: The derivative at a point a is defined as:
   • f′(a) = lim (f(a + h) - f(a)) / h as h approaches 0. Here, h is a very small number that represents a change in x.
2. Step 1: Write the Difference Quotient: First, substitute f(x) into the definition:
   • f′(a) = lim ((a + h)^2 - a^2) / h.
3. Step 2: Expand the Numerator: Expanding (a + h)^2 results in a^2 + 2ah + h^2, so:
   • f′(a) = lim (a^2 + 2ah + h^2 - a^2) / h, which simplifies to lim (2ah + h^2) / h.
4. Step 3: Simplify the Expression: Factor out h from the numerator:
   • f′(a) = lim h(2a + h) / h. The h cancels out:
   • f′(a) = lim (2a + h) as h approaches 0. This simplifies to 2a.
5. Conclusion: Thus, we find that the derivative of f(x) = x^2 is f′(x) = 2x.

Picture a ball rolling down a hill; you want to find out how steep the hill is at a certain point (the slope). To do this, you first observe just a tiny section of the hill right around that point. By analyzing this small segment (with h being a tiny movement up or down the hill), you determine that the slope at that point is how fast the ball is rolling at that exact moment. This captures the essence of determining the derivative: finding the slope right where you are, even with a slight movement to better understand the curve.

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