How do you prove derivatives of trig functions?

Unlocking the Secrets: How to Prove Derivatives of Trig Functions (Without the Headache)

Trig functions. We love ’em, we use ’em, but let’s be honest, sometimes they feel like they’re plotting against us. Especially when derivatives enter the chat. But fear not! Understanding how to actually prove those derivative rules? That’s where the real power lies. This isn’t just about memorizing formulas; it’s about truly grasping the why behind them. So, let’s break it down in a way that (hopefully) won’t make your brain explode.

The Toolkit: What You’ll Need

Think of this as prepping your workbench. Before we start hammering away at these proofs, we need a few essential tools. Consider them the nuts and bolts of trigonometric calculus:

• The Definition of a Derivative: This is ground zero, the fundamental concept. Remember that whole “limit as h approaches zero” thing? Yeah, that’s it:

  f'(x) = lim (h→0) f(x + h) – f(x) / h

  It looks intimidating, but it’s just a fancy way of saying “the instantaneous rate of change.”

• Limit Laws: These are your get-out-of-jail-free cards when dealing with limits. They let you split ’em up, move things around, and generally make your life easier.

• Trig Identities: Your Secret Weapons: These are crucial. Seriously, you can’t do this without knowing your trig identities inside and out. The biggies we’ll use are:

  • sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
  • cos(A + B) = cos(A)cos(B) – sin(A)sin(B)
  • sin²(x) + cos²(x) = 1 (This one’s like the Pythagorean theorem of trig)

• Special Trig Limits: The Magic Ingredients: These are the weird, almost mystical limits that make the whole thing work. Memorize these, tattoo them on your arm, do whatever it takes:

  • lim (x→0) sin(x) / x = 1
  • lim (x→0) (1 – cos(x)) / x = 0

  Trust me, they’re more useful than you think.

1. Sine’s Story: Proving d/dx sin(x) = cos(x)

Okay, let’s tackle the big one first. The derivative of sine is cosine. But why? Let’s see:

Boom. d/dx sin(x) = cos(x). Feels good, doesn’t it?

2. Cosine’s Counterpart: Showing d/dx cos(x) = -sin(x)

Now for cosine. The process is very similar to sine, so we can reuse some of the same tricks:

And there you have it: d/dx cos(x) = -sin(x). Notice that negative sign! Don’t forget it.

3. Tangent Takes the Stage: Proving d/dx tan(x) = sec²(x)

Tangent is where things get a little more interesting. We’ll use the quotient rule here, which is just a fancy way of saying “derivative of a fraction.”

Therefore, d/dx tan(x) = sec²(x). Not so bad, right?

4. Cotangent: Tangent’s Less Popular Cousin: Showing d/dx cot(x) = -csc²(x)

Cotangent is similar to tangent, just flipped. So, the proof is also similar:

Therefore, d/dx cot(x) = -csc²(x).

5. Secant: The Wildcard: Proving d/dx sec(x) = sec(x)tan(x)

Secant’s derivative is a bit of a weird one, but we can handle it.

Therefore, d/dx sec(x) = sec(x)tan(x).

6. Cosecant: The Grand Finale: Showing d/dx csc(x) = -csc(x)cot(x)

Last but not least, cosecant.

Therefore, d/dx csc(x) = -csc(x)cot(x).

The Ultimate Cheat Sheet: Trig Derivative Edition

FunctionDerivativesin(x)cos(x)cos(x)-sin(x)tan(x)sec²(x)cot(x)-csc²(x)sec(x)sec(x)tan(x)csc(x)-csc(x)cot(x)