Table of contents
- 0. Functions7h 52m
- Introduction to Functions16m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms34m
- Exponential & Logarithmic Equations35m
- Introduction to Trigonometric Functions38m
- Graphs of Trigonometric Functions44m
- Trigonometric Identities47m
- Inverse Trigonometric Functions48m
- 1. Limits and Continuity2h 2m
- 2. Intro to Derivatives1h 33m
- 3. Techniques of Differentiation3h 18m
- 4. Applications of Derivatives2h 38m
- 5. Graphical Applications of Derivatives6h 2m
- 6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
- 7. Antiderivatives & Indefinite Integrals1h 26m
- 8. Definite Integrals4h 44m
- 9. Graphical Applications of Integrals2h 27m
- 10. Physics Applications of Integrals 2h 22m
3. Techniques of Differentiation
Basic Rules of Differentiation
Problem 63b
Textbook Question
Let f(x) = x2 - 6x + 5.
Find the values of x for which the slope of the curve y = f(x) is 2.

1
Step 1: To find the slope of the curve y = f(x), we need to find the derivative of f(x). The function given is f(x) = x^2 - 6x + 5.
Step 2: Differentiate f(x) with respect to x. The derivative, f'(x), represents the slope of the curve at any point x. Use the power rule: if f(x) = ax^n, then f'(x) = n*ax^(n-1).
Step 3: Apply the power rule to each term in f(x). The derivative of x^2 is 2x, the derivative of -6x is -6, and the derivative of a constant (5) is 0. Therefore, f'(x) = 2x - 6.
Step 4: Set the derivative equal to the given slope value. We want the slope to be 2, so set f'(x) = 2x - 6 equal to 2.
Step 5: Solve the equation 2x - 6 = 2 for x. This will give you the x-values where the slope of the curve is 2.

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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures the rate at which the function's value changes as its input changes. It is often interpreted as the slope of the tangent line to the curve at a given point. For the function f(x) = x² - 6x + 5, finding the derivative f'(x) will allow us to determine the slope of the curve at any point x.
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Slope of the Curve
The slope of the curve at a specific point is given by the value of the derivative at that point. In this problem, we are interested in finding the values of x where the slope of the curve, represented by f'(x), equals 2. This involves setting the derivative equal to 2 and solving for x.
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Quadratic Functions
A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax² + bx + c. The graph of a quadratic function is a parabola. In this case, the function f(x) = x² - 6x + 5 is a quadratic function, and understanding its properties, such as its vertex and axis of symmetry, can provide insights into its behavior and the solutions to the slope problem.
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