Section 3.3: Derivative of Trigonometric Functions:

1.  Differentiate the following functions.   solution

(i) \(y=3\sin(\theta) - 4e^{\theta}\, \tan (\theta) \)     (ii) \(y=\sec(\theta) (2\theta - 4 \cos(\theta)) \)   

2.   Differentiate the following functions.   solution

(i) \(f(x)=\dfrac{2x}{x-\cot x} \)     (ii) \(y= 5 t\cos t \tan t \)

3. If   \( f(x) = e^x \sec x \), then find \( f'(0)\).    solution

4. Differentiate:   (a) \( f(x)= x \cos x \)   solution     (b) \(\displaystyle{f(x)= \frac{\tan x}{x^2} }\),     solution

5.  If   \( f (\theta)= \theta \cos \theta \),  find the value of \( f'(0) \).     solution

6.  If   \( f(\pi/3)= 4,   f'(\pi/3)=-2\)  and   \(g(x)=f(x) \sin x \), then find \( g'(\pi/3)\).    solution

7. Find an equation of the tangent line to the curve  \(y= x-\cos x \)  at  \( x=0\).    solution

8. Find an equation of the tangent line to the curve at the given point.   solution   \[y=2\sec (x)-4 \cos (x), \quad P=\left(\frac{\pi}{3}, 2\right) \]

9.  If   \( f(x) = \sec x \), then find \(f''(\pi/4)\).    solution

10.   Suppose \(f(\pi / 3)= 6\) and \(f'(\pi / 3)=-5,\) and let \(g(x)=f(x) \sin (x)\) and \(h(x)=\cos (x) / f(x) \). Find (a) \(g'(\pi/3) \) and (b) \(h'(\pi/3) \).

solution