2. Derivatives of Exponential, Logarithmic, and Trigonometric Functions
  3. Derivatives of Tangent and Reciprocal Trigonometric Functions

Exercises


  1. Determine \(\dfrac{dy}{dx}\).
    1. \(y = \tan(5x+2)\)
    2. \(y = \sec(2x)\)
    3. \(y = \cot(\sqrt{x})\)
    4. \(y = \csc^{2}(x)\)
  2. Determine \(f'(x)\).
    1. \(f(x) = \Big(\tan(x)+\sin(x)\Big)^{2}\)
    2. \(f(x) = \sec(x)\tan(x)\)
    3. \(f(x) = \ln\Big(\sec(x)\Big)\)
    4. \(f(x) = \cot^{3}(3x)\)
    1. Determine the value of \(f'(0)\) if \(f(x) = 5\tan(x) - \tan(3x)\).
    2. Determine the value of \(g'\left(\frac{\pi}{4}\right)\) if \(g(x) = \csc(x)\).
    3. Determine the value of \(s'(0)\) if \(s(t) = t \sec(t)\).
    4. Determine the value of \(h'(\pi)\) if \(h(x) = \tan^{2}(\cos(x))\).
  4. Find all values of \(x\) in the interval \(\left[0,2\pi\right]\) for which the slope of the tangent to \(y=f(x)\) is zero.
    1. \(y = e^{x}\csc(x)\)
    2. \(y = \cos(x) \cot(x)\)
  5. Determine the first and second derivative of the given function.
    1. \(f(x) = \tan^{2}(x)\)
    2. \(f(x) = \sec(x)+\tan(x)\)
  6. Let \(y = \sec(x)\csc(x)\).
    1. Determine \(\dfrac{dy}{dx}\) in terms of only the functions \(\sin(x)\) and \(\cos(x)\).
    2. Find the equations of all horizontal tangents to \(y=f(x)\).
    3. Find an equation of the tangent line to \(y=f(x)\) at the point with \(x\)-value \(\dfrac{\pi}{3}\).
    1. For what values of \(x\) in the interval \((0,2\pi)\) is the function \(f(x) = \sin(x) \tan(x)\) increasing?
    2. For what values of \(x\) in the interval \((0,2\pi)\) is the function \(f(x) = \tan^{2}(x)\) concave upward?
  8. Show that \(\dfrac{d}{dx}\left(~\ln\!\sqrt{\dfrac{\tan^{2}(x)-1}{\sec^{2}(x)}}~~\right) =- \tan(2x)\).