3.5 - Trig Derivatives

#calc/concept

The derivatives

${\displaystyle \frac{d}{dx}\sin x=\cos x}$

${\displaystyle \frac{d}{dx}\cos x=-\sin x}$

${\displaystyle \frac{d}{dx}\tan x={\sec }^{2}x}$

${\displaystyle \frac{d}{dx}\cot x=-{\csc }^{2}x}$

${\displaystyle \frac{d}{dx}\sec x=\sec x\tan x}$

${\displaystyle \frac{d}{dx}\csc x=-\csc x\cot x}$

• Some proofs can be found here.
  • To derive the others, use things like the product and quotient rules, the quotient trig identities, and already-found identities.

Periodicity

• Sine and cosine derivatives are periodic – this means that they repeat every certain number of powers.
  • The first and fifth derivatives are the same, the second and sixth, etc…
    $f(x)=\sin x$
    $f^{\prime }(x)=\cos x$
    $f^{″}(x)=-\sin x$
    $f^{‴}(x)=-\cos x$
    $f^{⁗}(x)=\sin x$
    $f^{\prime \prime \prime \prime \prime }(x)=\cos x$

Pythagorean Identities

(These aren’t really covered but are important to know anyway.)

${\sin }^{2}x+{\cos }^{2}x=1$

${\tan }^{2}x+1={\sec }^{2}x$

${\csc }^{2}x+1={\cot }^{2}x$