Chapter 3 Section 3: Unveiling Trigonometric Derivatives

Welcome back to Professor Baker's Math Class! In this section, we'll be exploring the fascinating world of trigonometric derivatives. Get ready to expand your calculus toolkit and tackle new challenges!

Why are trigonometric derivatives important? Trigonometric functions model cyclical phenomena, from oscillations in physics to economic trends. Understanding their derivatives allows us to analyze rates of change within these dynamic systems.

The Core Trigonometric Derivatives

Let's start with the fundamental derivatives:

  • The derivative of sine: If $f(x) = sin(x)$, then $f'(x) = cos(x)$.
  • The derivative of cosine: If $f(x) = cos(x)$, then $f'(x) = -sin(x)$.

These two form the basis for deriving the rest!

Deriving the Other Trig Derivatives

We can use the quotient rule to find the derivatives of the remaining trigonometric functions:

  • Tangent: Since $tan(x) = \frac{sin(x)}{cos(x)}$, we can apply the quotient rule. The result is $ \frac{d}{dx} tan(x) = sec^2(x)$. That is, if $f(x) = tan(x)$, then $f'(x) = sec^2(x)$.
  • Cotangent: Since $cot(x) = \frac{cos(x)}{sin(x)}$, we can apply the quotient rule. The result is $ \frac{d}{dx} cot(x) = -csc^2(x)$. That is, if $f(x) = cot(x)$, then $f'(x) = -csc^2(x)$.
  • Secant: Since $sec(x) = \frac{1}{cos(x)}$, we can apply the quotient rule or the chain rule. The result is $ \frac{d}{dx} sec(x) = sec(x)tan(x)$. That is, if $f(x) = sec(x)$, then $f'(x) = sec(x)tan(x)$.
  • Cosecant: Since $csc(x) = \frac{1}{sin(x)}$, we can apply the quotient rule or the chain rule. The result is $ \frac{d}{dx} csc(x) = -csc(x)cot(x)$. That is, if $f(x) = csc(x)$, then $f'(x) = -csc(x)cot(x)$.

Key Takeaways and Tips

  • Memorization is Key: Knowing the derivatives of $sin(x)$ and $cos(x)$ is crucial. From there, you can derive the rest.
  • Quotient Rule is Your Friend: The quotient rule is a powerful tool for deriving the derivatives of $tan(x)$, $cot(x)$, $sec(x)$, and $csc(x)$.
  • Practice, Practice, Practice: Work through plenty of examples to solidify your understanding.

Examples

Let's look at a quick example: Find the derivative of $y = 3sin(x) + 2cos(x)$.

Solution: Using the sum and constant multiple rules, we have:

$$y' = 3\frac{d}{dx}sin(x) + 2\frac{d}{dx}cos(x) = 3cos(x) - 2sin(x)$$

Next Steps

Now that you've learned the basics of trigonometric derivatives, it's time to practice applying them in various problems. Don't hesitate to review the powerpoint provided and rewatch the video to reinforce your understanding. Good luck, and keep exploring the amazing world of calculus!