Derivative by increment method (By definition with limit)
Summary
TLDRIn this video, the concept of calculating the derivative of the function f(x) = 2x + 1 is explained using the increment method, also known as the definition of a derivative via limits. The process involves calculating f(x + delta x), simplifying the expression, and finding the difference between f(x + delta x) and f(x). After simplifying the fraction, the limit as delta x approaches zero is taken, leading to the derivative. The video concludes with a similar exercise for viewers to practice and encourages engagement through likes, subscriptions, and comments.
Takeaways
- 😀 The derivative of a function can be calculated using the method of increments, which involves applying the limit definition of the derivative.
- 😀 The formula for the derivative is: f'(x) = lim(Δx → 0) [(f(x + Δx) - f(x)) / Δx].
- 😀 To begin, substitute x with (x + Δx) in the original function to calculate f(x + Δx).
- 😀 After substitution, simplify the expression for f(x + Δx) by distributing and combining like terms.
- 😀 Once f(x + Δx) is calculated, subtract f(x) from it to find the numerator in the derivative formula.
- 😀 Cancel out any common terms between f(x + Δx) and f(x) to simplify the subtraction process.
- 😀 The remaining terms in the numerator after cancellation represent the change in the function.
- 😀 The next step is to divide the result by Δx, and then simplify the expression.
- 😀 The final step is to compute the limit as Δx approaches 0. If no terms involving Δx remain, the limit is simply the constant value left.
- 😀 In this specific example, the derivative of the function f(x) = 2x + 1 is 2, since after simplification, no Δx remains in the limit expression.
Q & A
What is the main topic of the video?
-The main topic of the video is calculating the derivative of a function using the method of increments, which is based on the definition of the derivative as a limit.
What is the formula for calculating the derivative in this video?
-The formula for the derivative is: f'(x) = lim(Δx → 0) [(f(x + Δx) - f(x)) / Δx].
How does the presenter simplify the derivative formula?
-The presenter simplifies the formula by applying it to a specific function f(x) = 2x + 1, and shows step-by-step how to calculate f(x + Δx) and cancel out common terms.
What is the first step in applying the formula?
-The first step is to calculate f(x + Δx) by substituting (x + Δx) into the function f(x). In this case, f(x) = 2x + 1, so f(x + Δx) becomes 2(x + Δx) + 1.
What happens after calculating f(x + Δx)?
-After calculating f(x + Δx), the next step is to subtract f(x) from f(x + Δx), cancel out the common terms (like 2x and +1), and simplify the expression.
What is the result after performing the subtraction of f(x + Δx) and f(x)?
-The result of the subtraction is 2Δx, which is the difference between f(x + Δx) and f(x) after canceling out the common terms.
What is the next step after getting the result of the subtraction?
-The next step is to divide the result of the subtraction (2Δx) by Δx, which gives 2. Then, the limit as Δx approaches 0 is calculated.
What does the limit of 2Δx / Δx as Δx approaches 0 yield?
-The limit of 2Δx / Δx as Δx approaches 0 simplifies to 2, because the Δx terms cancel out, leaving the constant 2.
What is the final result of the derivative calculation?
-The final result of the derivative calculation is 2, which is the value of f'(x), the derivative of f(x) = 2x + 1.
What example does the presenter give for viewers to practice after the video?
-The presenter gives the example of f(x) = 10x - 9 for viewers to calculate the derivative using the same method demonstrated in the video.
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