The derivative of sine from the limit definition

1. Formula
   Observation

   On the left side of the screen, $f(x) = \sin x$ is written.

f(x) = \sin x

The trigonometric function to be differentiated

Real numbers

1. Formula
   Observation

   $h$ appears in the derivative definition formula, and the limit process is $h \to 0$.

h

Increment of the independent variable

Real number approaching 0

1. Formula
   Observation

   Variables $A$ and $B$ are used in the sum-to-product formula.

A, B

Arbitrary angles

Real numbers

1. Formula
   Observation

   f(x) = \sin x

f(x)

The function to be differentiated, here the sine function

Real numbers

1. Formula
   Observation

   h \to 0

h

Increment of the independent variable

Real numbers, approaching 0

1. Formula
   Observation

   \lim_{h \to 0}

\lim

Limit symbol

Calculus

1. Formula
   Observation

   \sin A - \sin B

\sin

Sine function

Real numbers

1. Formula
   Observation

   \cos \frac{A+B}{2}

\cos

Cosine function

Real numbers

1. Formula
   Observation

   The top-left corner of the screen displays f(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h}

f(x)

The value of function f at x, here referring to the definition of the derivative of the sine function \sin x

Real numbers

1. Formula
   Observation

   The denominator and below the limit symbol show h and h \to 0

h

The increment of the independent variable, approaching 0

Non-zero real numbers

1. Formula
   Observation

   Inside the parentheses of the sine function are x and x+h

x

Independent variable

Real numbers

1. Audio
   Observation

   Narration paraphrase: The derivative of a function at a point is defined as the limit of the ratio of the change in the function value to the change in the independent variable as the latter approaches zero.

2. Formula
   Observation

   On the right side of the screen, $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ is written.

The derivative of a function at a point is defined as the limit of the ratio of the change in the function value to the change in the independent variable as the latter approaches zero.

1. The limit exists

2. Editorial scope for this derivative calculation: angles are in radians; x is fixed and real, nonzero h approaches0. The standard sine-ratio limit and cosine continuity are assumed prerequisites, not proved by this video.

1. Audio
   Observation

   Narration paraphrase: Converts the difference of two sine functions into a product of sine and cosine functions.

2. Formula
   Observation

   At the bottom of the screen, $\sin A - \sin B = 2\sin\frac{A-B}{2}\cos\frac{A+B}{2}$ is written.

Converts the difference of two sine functions into a product of sine and cosine functions.

1. Editorial scope for this derivative calculation: angles are in radians; x is fixed and real, nonzero h approaches0. The standard sine-ratio limit and cosine continuity are assumed prerequisites, not proved by this video.

1. Formula
   Observation

   f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

The derivative of a function at a point is defined as the limit of the ratio of the change in the function value to the change in the independent variable as the latter approaches zero.

1. The limit exists

2. Editorial scope for this derivative calculation: angles are in radians; x is fixed and real, nonzero h approaches0. The standard sine-ratio limit and cosine continuity are assumed prerequisites, not proved by this video.

1. Formula
   Observation

   \sin A - \sin B = 2\sin\frac{A-B}{2}\cos\frac{A+B}{2}

Converts the difference of two sine functions into a product of sine and cosine functions.

1. Editorial scope for this derivative calculation: angles are in radians; x is fixed and real, nonzero h approaches0. The standard sine-ratio limit and cosine continuity are assumed prerequisites, not proved by this video.

1. Formula
   Observation

   \lim_{x \to 0} \frac{\sin x}{x} = 1

As the independent variable approaches zero, the limit of the ratio of the sine function value to the variable is 1.

1. Does not hold in degrees; must be in radians

2. Editorial scope for this derivative calculation: angles are in radians; x is fixed and real, nonzero h approaches0. The standard sine-ratio limit and cosine continuity are assumed prerequisites, not proved by this video.

3. Editorial variable scope: x in this fundamental limit is a local varying dummy argument, distinct from the fixed point whose derivative is calculated.

1. Formula
   Observation

   The top-right corner of the screen displays \sin A - \sin B = 2\sin\frac{A-B}{2}\cos\frac{A+B}{2} and its substituted form

A trigonometric identity that converts the difference of two sine functions into a product of sine and cosine.

1. A, B are any real numbers

2. Editorial scope for this derivative calculation: angles are in radians; x is fixed and real, nonzero h approaches0. The standard sine-ratio limit and cosine continuity are assumed prerequisites, not proved by this video.

1. Formula
   Observation

   The middle-right side of the screen displays \lim_{x \to 0} \frac{\sin x}{x} = 1

When the independent variable approaches 0, the limit of the ratio of the sine value to the independent variable is 1.

1. Angle unit is radians

2. Editorial scope for this derivative calculation: angles are in radians; x is fixed and real, nonzero h approaches0. The standard sine-ratio limit and cosine continuity are assumed prerequisites, not proved by this video.

3. Editorial variable scope: x in this fundamental limit is a local varying dummy argument, distinct from the fixed point whose derivative is calculated.

1. Audio
   Observation

   Narration paraphrase: Transforms the numerator of the limit expression in the derivative definition into a product form, preparing for subsequent limit calculation.

2. Formula
   Observation

   In the lower-left corner of the screen, $f'(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h}$ is written, followed by the sum-to-product formula and the beginning of the expansion for $\sin(x+h) - \sin x$ after substitution.

1. The clip ends after substituting the sum-to-product formula; the complete simplification and limit evaluation process is not shown.

1. Expression

   $$f'(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h}$$
   Explanation

   Substitute $f(x) = \sin x$ into the definition of the derivative.

   Justification

   Definition of the derivative

   Shown in the video
2. Expression

   $$\sin(x+h) - \sin x = 2\sin\frac{(x+h)-x}{2}\cos\frac{(x+h)+x}{2}$$
   Explanation

   Apply the sum-to-product formula for sine to transform the numerator.

   Justification

   Sum-to-product formula

   Shown in the video

Transforms the numerator of the limit expression in the derivative definition into a product form, preparing for subsequent limit calculation.

1. Formula
   Observation

   Step-by-step derivation process

1. Expression

   $$f'(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h}$$
   Explanation

   Substitute into the definition of the derivative

   Justification

   Definition of the derivative

   Shown in the video
2. Expression

   $$= \lim_{h \to 0} \frac{2\sin\frac{x+h-x}{2}\cos\frac{x+h+x}{2}}{h}$$
   Explanation

   Use the sum-to-product formula to expand the numerator

   Justification

   Sum-to-product formula

   Shown in the video
3. Expression

   $$= \lim_{h \to 0} \frac{2\sin\frac{h}{2}\cos\frac{2x+h}{2}}{h}$$
   Explanation

   Simplify the expressions inside the parentheses

   Justification

   Algebraic simplification

   Shown in the video
4. Expression

   $$= \lim_{h \to 0} \frac{\sin\frac{h}{2}}{\frac{h}{2}} \cdot \cos\left(x + \frac{h}{2}\right)$$
   Explanation

   Split the denominator h into 2 * (h/2), cancel with the 2 in the numerator, and separate the cosine term

   Justification

   Algebraic manipulation

   Derived from the video

This analysis segment reaches the standard-limit form and ends before the final result \cos x; the actual full video continues and completes the derivation in the next segment.

1. Formula
   Observation

   The left side of the screen shows the complete algebraic transformation process from the definition of the derivative to the final result

2. Audio
   Observation

   Narration paraphrase: The derivative of the sine function is the cosine function, i.e., (\sin x)' = \cos x.

1. Editorial notation check: the small prime was missed in the received later-segment recognition. Clear earlier native derivative definitions and the complete mathematical argument verify that this expression is the derivative, not the function value; the received OCR is not claimed unambiguous.

1. Expression

   $$f'(x) = \lim_{h \to 0} \frac{\sin(x+h) - \sin x}{h}$$
   Explanation

   Write down the definition of the derivative for the sine function.

   Justification

   Limit definition of the derivative

   Supplementary explanation
2. Expression

   $$= \lim_{h \to 0} \frac{2\sin\frac{h}{2}\cos\frac{2x+h}{2}}{h}$$
   Explanation

   Expand the numerator using the sum-to-product formula.

   Justification

   Trigonometric identity \sin A - \sin B = 2\sin\frac{A-B}{2}\cos\frac{A+B}{2}

   Shown in the video
3. Expression

   $$= \lim_{h \to 0} \frac{\sin\frac{h}{2}}{\frac{h}{2}} \cdot \lim_{h \to 0} \cos\frac{2x+h}{2}$$
   Explanation

   Move the constant 2 into the denominator to form \frac{h}{2}, and split the expression using the product rule for limits.

   Justification

   Algebraic manipulation and limit operation rules

   Shown in the video
4. Expression

   $$= 1 \cdot \cos x$$
   Explanation

   The first limit equals 1 according to the important limit; the second limit yields \cos x by direct substitution of h=0 due to the continuity of the cosine function.

   Justification

   Important limit \lim_{t \to 0} \frac{\sin t}{t} = 1 and the continuity of the cosine function

   Shown in the video
5. Expression

   $$(\sin x)' = \cos x$$
   Explanation

   State the final conclusion.

   Justification

   Result of the derivation

   Shown in the video

The derivative of the sine function is the cosine function, i.e., (\sin x)' = \cos x.

1. Diagram
   Observation

   In the upper-right corner of the screen, a function graph is displayed, including the curve $y=f(x)$, a secant line, a tangent line, and related coordinate and length annotations.

1. Curve y=f(x)

2. Point P(x, f(x))

3. Point Q(z, f(z))

4. Secant line

5. Tangent line

6. Horizontal segment h = z - x

7. Vertical segment f(z) - f(x)

1. No dynamic changes; it is a static display image.

1. Secant slope formula Secant slope is \frac{f(z) - f(x)}{z - x}

2. Derivative definition formula Derivative of f at x is f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{z \to x} \frac{f(z) - f(x)}{z - x}

Intuitively demonstrates the geometric meaning of the derivative as the slope of the tangent line, and the relationship between the limit of the secant slope and the definition of the derivative.

1. Animation
   Observation

   Handwritten formulas appear step by step on grid paper

1. Red handwriting

2. Blue handwriting

3. Grid paper

1. Formulas are written from left to right, top to bottom

1. Background grid paper remains unchanged

Demonstrates the derivation process of the derivative of the sine function.

1. Animation
   Observation

   A red box sequentially circles x in \sin x, the denominator x, \frac{h}{2} in \sin\frac{h}{2}, and the denominator \frac{h}{2}

1. Red box

2. Variables in mathematical formulas

1. The red box moves between different variables

1. The algebraic structure of the formula itself remains unchanged

Emphasizes that when using the important limit, the variable inside the sine function must be exactly identical to the variable in the denominator, and both must approach 0.

1. Audio
   Observation

   Narration paraphrase: In the process of finding limits, when variables approach a certain value and both numerator and denominator approach 0, this is a 0/0 indeterminate form, which needs to be solved through simplification (such as using important limits).

Believing that a limit expression with a denominator of 0 is meaningless.

In the process of finding limits, when variables approach a certain value and both numerator and denominator approach 0, this is a 0/0 indeterminate form, which needs to be solved through simplification (such as using important limits).

1. Audio
   Observation

   Narration paraphrase: It must be ensured that the entire expression inside the parentheses of the sine function is exactly identical to the denominator, and that this entire expression approaches 0.

2. Animation
   Observation

   Uses a red box to highlight the same variables in the numerator and denominator

Believing that as long as the numerator is sin(some expression) and the denominator is another expression, one can directly apply the conclusion that the limit is 1.

It must be ensured that the entire expression inside the parentheses of the sine function is exactly identical to the denominator, and that this entire expression approaches 0.

1. Audio
   Observation

   Narration paraphrase: Applies the general definition of the derivative to the specific function $f(x) = \sin x$, constructing the limit expression for differentiation.

Applies the general definition of the derivative to the specific function $f(x) = \sin x$, constructing the limit expression for differentiation.

1. Audio
   Observation

   Narration paraphrase: In the sum-to-product formula, let $A = x+h$ and $B = x$, used to simplify the numerator of the derivative limit expression.

In the sum-to-product formula, let $A = x+h$ and $B = x$, used to simplify the numerator of the derivative limit expression.

1. Formula
   Observation

   Using the definition of the derivative for derivation

The definition of the derivative is applied to derive the derivative of the sine function.

1. Formula
   Observation

   Using the sum-to-product formula

The sum-to-product formula is used to simplify the numerator in the definition of the derivative.

1. Formula
   Observation

   Mentioning the important limit

The important limit is the key tool for solving the 0/0 type limit that appears during the derivation process.

1. Formula
   Observation

   The second step of the derivation directly uses the sum-to-product formula given in the top-right corner

The sum-to-product formula is applied to the transformation of the numerator in the definition of the derivative.

1. Formula
   Observation

   In the fourth step of the derivation, calculating the first limit directly yields 1

The important limit is used to calculate the limit value of the first factor after splitting.

1. Audio
   Observation

   Narration paraphrase: How to find the derivative of sin x using the definition of the derivative?

1. Definition of the Derivative
2. Initial Derivation of the Derivative of sin x Using the Definition

1. Formula
   Observation

   The screen displays the sum-to-product formula for sin A - sin B.

1. Sum-to-Product Formula for Sine

1. Audio
   Observation

   Narration paraphrase: How to find the derivative of sin(x) using the definition?

1. Definition of the Derivative
2. Sum-to-Product Formula
3. Important Limit
4. Derivation of the Derivative of the Sine Function

1. Audio
   Observation

   Narration paraphrase: How to prove that the derivative of sin x is cos x using the limit definition?

1. Derivation of the derivative of the sine function

1. Animation
   Observation

   The red box specifically circles the variables that need to match

1. Variable matching in the important limit
2. Important limit