3.4: Derivatives of Trigonometric Functions

1. Given $y = \sin x + \cos x + 2x^3$, find $\dfrac{dy}{dx}$.
    
2. Given $f(x) = 2x\sec x$, find $\dfrac{df}{dx}$.
    
3. Given $g(x) = \dfrac{\tan x}{x}$, find $\dfrac{dg}{dx}$.
    
4. Given $h(x) = \csc x \cot x$, find $\dfrac{dh}{dx}$.
    
5. Given $y = 4\sin x - 3\sqrt{x}$, find $y'$.
    
6. Given $f(x) = x\cos^2 x$, find $f'(x)$.
    
7. Given $g(x) = \dfrac{\sec x}{\csc x}$, find $g'(x)$.
    
8. Given $h(x) = \dfrac{(x + 1)\tan x}{1 + \cot x}$, find $h'(x)$.
    
9. Given $r = \dfrac{\sec \theta}{1 + \tan \theta}$, find $\dfrac{dr}{d\theta}$.
    
10. Given $x = \sin^2 t + \cos^2 t$, find $\dfrac{dx}{dt}$.
     
11. Given $f(\theta) = -3(\sec^2 \theta - \tan^2 \theta)$, find $\dfrac{df}{d\theta}$.
     
12. Given $g(x) = \dfrac{1 + x\tan x}{\sin x \sec x}$, find $\dfrac{dg}{dx}$.
     
13. Given $h(x) = \dfrac{2 - \cos x \csc x}{(x^2 + 1)\cot x}$, find $\dfrac{dh}{dx}$.
     
14. Given $y = 3x\sin x$, find $y'$ and $y''$.
     
15. Given $f(x) = 2\sec x - 5\tan x$, find $\dfrac{df}{dx}$ and $\dfrac{d^2f}{dx^2}$.
     
16. Find the equation of the line tangent to the graph of $g(x) = \sin x$ at $x = \pi$.
     
17. Find the equation of the line tangent to the graph of $h(x) = 3\csc x$ at $x = \dfrac{\pi}{2}$.
     
18. Find the equation of the line tangent to the graph of $y = \sec x + \tan x$ at $x = -\dfrac{\pi}{4}$.