How can any of these be true?

Calculus Level 3

Which of the following equations is true?

\frac d{dx} ( \sin x \cdot \sin x ) = \sin(x + x)

\frac d{dx} ( \tan x \cdot \tan x ) = \tan(x + x)

\frac d{dx} ( \cos x \cdot \cos x ) = \cos (x + x)

1 solution

Rahil Sehgal
Apr 3, 2017

We can directly apply chain rule to get our answer.

$\dfrac{d}{dx} ( \sin x \cdot \sin x ) = ( \cos x \cdot \sin x ) + ( \sin x \cdot \cos x )$ which is equal to $\sin (x+x)$

Yeah, for completeness, you should also show why the other 2 options are wrong.

And as a bonus: Can you prove that $\dfrac d{dx} \sin x = \cos x$ and $\dfrac d {dx} \cos x = -\sin x$ ?

Pi Han Goh - 4 years, 2 months ago

I have added a proof that $\dfrac{d}{dx} \sin x = \cos x$ in my solution. Hope that clarifies... :)

Rahil Sehgal - 4 years, 2 months ago

Good attempt, but this leaves us with:

How do you know that $\displaystyle \lim_{x\to0} \dfrac{ \cos x - 1}{x } = 0$ and $\displaystyle \lim_{x\to0} \dfrac{ \sin x}{x } =1$ ?

Pi Han Goh - 4 years, 2 months ago

@Pi Han Goh – We can prove it by L'Hospital's Rule.

Rahil Sehgal - 4 years, 2 months ago

@Rahil Sehgal – No, that's a circular argument. My question here was: "Can you prove it X?" Then you said, "yeah, using Y. "

But if you want to prove Y is true, you cannot just say "Yeah, use X again".

Do you understand the fallacy that you've committed?

Pi Han Goh - 4 years, 2 months ago