Problems

01

Trig derivatives - by the rules

Apply the quotient rule and the facts that $\frac{d}{dx}\sin x=\cos x$ and $\frac{d}{dx}\cos x=-\sin x$ to find the derivatives of these functions:

• (a) $\frac{d}{d\theta }\tan \theta$
• (b) $\frac{d}{d\theta }\sec \theta$
• (c) $\frac{d}{d\theta }\csc \theta$
• (d) $\frac{d}{d\theta }\cot \theta$

02

Derivatives - composite functions with trig

Differentiate the following functions:

• (a) $y={\sin }^2\theta$
• (b) $f(\theta )={\sec }^2(\cos (\theta ))$
• (c) $g(t)=\frac{1}{\sin 2t+e^{3t}}$
• (d) $y(\theta )=e^{\tan \theta }$
• (e) $h(x)=\tan (e^{1/x})$

03

Trig derivatives

Differentiate the following functions and simplify your answers:

• (a) $f(x)=3\sec x+4\cot x$
• (b) $f(\theta )=\frac{\tan \theta }{1+\tan \theta }$
• (c) $g(x)=\frac{2x\sin x}{2+\cos x}$
• (d) $y=\theta \sin \theta \cos \theta$

04

First, second, third derivatives

Suppose $f(x)=3x\cos (x^2)$.

• (a) Compute $f^{\prime }(x)$.
• (b) Compute $f^{\prime \prime }(x)$.
• (c) Compute $f^{\prime \prime \prime }(x)$.

05

Tangent line to composite trig function

Find the equation of the tangent line to the graph of $f(x)=\sin (\sin x)$ when $x=\pi$.