The video “Calculus 1 Lecture 2.5: Finding Derivatives of Trigonometric Functions” by Professor Leonard explains how to derive and use the formulas for the derivatives of the six basic trigonometric functions: sine, cosine, tangent, secant, cosecant, and cotangent.
Explanation#
1. Why Learn Trig Derivatives?#
- Trigonometric derivatives are used throughout calculus—especially in more advanced topics like substitution, solving differential equations, and solving real-world physics problems.
- Memorizing the derivatives for all six trig functions (sine, cosine, tangent, cotangent, secant, cosecant) is crucial.
2. Proof of the Derivative of Sine#
- The video shows how the derivative of $\sin x$ is found using the limit definition of the derivative:
$$ \frac{d}{dx}[\sin x] = \lim_{h\to 0} \frac{\sin(x+h) - \sin(x)}{h} $$
- The sine addition formula is used: $\sin(x + h) = \sin x \cos h + \cos x \sin h$.
- After algebraic manipulation, limits, and trig limits ($\lim_{h \to 0} \sin h / h = 1$, $\lim_{h \to 0} (1 - \cos h)/h = 0$), the result is:
$$ \frac{d}{dx}[\sin x] = \cos x $$
- Similarly, the derivative of $\cos x$ is $-\sin x$.
3. Derivatives of All Six Trig Functions#
- The derivatives of the other trig functions are derived using quotient and product rules, since:
- $\tan x = \frac{\sin x}{\cos x}$ and so on.
- The “memorization table” you need is:
- $\frac{d}{dx}[\sin x] = \cos x$
- $\frac{d}{dx}[\cos x] = -\sin x$
- $\frac{d}{dx}[\tan x] = \sec^2 x$
- $\frac{d}{dx}[\cot x] = -\csc^2 x$
- $\frac{d}{dx}[\sec x] = \sec x \tan x$
- $\frac{d}{dx}[\csc x] = -\csc x \cot x$
4. Examples Using Product and Quotient Rules#
- The video applies these formulas with the product rule (e.g., $y = x \sin x$), emphasizing recognizing when to use it.
- For $y = x \sin x$: $y’ = x \cos x + \sin x$
- When using quotient rule (e.g., derivatives involving ratios like $\frac{\sin x}{1 + \cos x}$), careful use of parentheses and step-by-step differentiation is demonstrated.
5. Higher-Order Derivatives and Oscillation#
- Repeatedly differentiating trigonometric functions (e.g., taking the second, third, fourth, etc., derivative of $\sin x$) results in a cyclical pattern due to the periodic nature of trig functions.
- This relates to physical phenomena like oscillations (springs, pendulums), where derivatives describe velocity and acceleration in terms of position.
6. Physical Application Example#
- A spring/mass system oscillating is modeled as a cosine or sine function.
- The derivative gives the velocity, and zeros of the derivative indicate turning points (maximum extension or compression of the spring).
7. Importance of Chain Rule#
- The video concludes by explaining the need for another differentiation technique, the chain rule, for handling compositions like $\sin(x^2)$ or more complex situations—this is presented as the next step in calculus after mastering trig, product, and quotient rules.
Summary#
This lecture teaches how to find and use the derivatives of trigonometric functions. It shows:
- The proofs using the limit definition and trig identities.
- Memorization and application of all six derivatives.
- How to use these derivatives in combination with product and quotient rules.
- How derivatives cycle back to the function, reflecting periodic behavior.
- The application to real-world oscillating systems and the transition into needing the chain rule.
Understanding and being able to use these derivatives is essential for success in calculus and in solving real-world and exam problems involving trigonometric functions.
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