Table of contents

0. Functions7h 52m
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
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

3. Techniques of Differentiation

Basic Rules of Differentiation

Calculus

3. Techniques of Differentiation

Basic Rules of Differentiation

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2:25 minutes

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.

Verified step by step guidance

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.

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.

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.