Trigonometry, the Chain Rule
& Implicit Differentiation

  1. Determine a formula for the derivative of each of the following explicitly-defined functions.

    \(\displaystyle a(x) = \sqrt{x^3}-\pi\tan(x) \)
    \(\displaystyle b(x) = x^5\sin(x) \)
    \(\displaystyle c(x) = \sqrt{\sec(x)} \)
    \(\displaystyle f(x) = \tfrac{3x}{\cos(x)} \)
    \(\displaystyle g(x) = \csc\bigl(x^3+x^2+x+1\bigr) \)
    \(\displaystyle h(t) = \sec\bigl(\sqrt{7t}\bigr) \)

  2. For these implicitly defined functions \(y = f(x)\), determine a formula for \(\frac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(x\) and \(y.\) Remember that you can also denote \(\frac{\mathrm{d}y}{\mathrm{d}x}\) as either \(y'\) or as \(\dot y.\)

    \(\displaystyle xy + y^7 + x^2 = 5 \)
    \(\displaystyle \sin(x)\cos(y) = \tan(xy) \)
    \(\displaystyle (xy)^3-2x(x+1)-y=2\)
  3. What is an equation for the line tangent to the graph of the function \({h(t) = \sec\bigl(\sqrt{7t}\bigr)}\) at the point where \(t = 2?\)
  4. Verify that the point \((1,2)\) lies on the curve defined implicitly by the equation \((xy)^3-2x(x+1)-y=2,\) then find an equation of the line tangent to the curve at that point.
  5. What’s an equation of the line tangent to the curve defined implicitly by the equation \(\tan(xy) = 1\) at the point \(\bigl(\frac{\pi}{2}, \frac{1}{2}\bigr)?\)
  6. What is the \(307^\text{th}\) derivative of sine?
  7. What is the \(13^\text{th}\) derivative of \(\cos\bigl(2x\bigr)?\)
  8. Jungic, Menz, Pyke Let \(F(x) = f\bigl(g(x)\bigr).\) Given the following information, what is \(F'(1)\)?
    \(g(1) = 2\)
    \(g'(1) = 3\)
    \(f(1) = 5\)
    \(f'(1) = 7\)
    \(f(2) = 11\)
    \(f'(2) = 1\)
  9. The quotient rule \[ \frac{\mathrm{d}}{\mathrm{d}x} \Bigl(\tfrac{f}{g}\Bigr) = \frac{f'g-fg'}{g^2} \] is superfluous; it’s simply a combination of the power rule, product rule, and chain rule. Prove this by noticing that \(f/g = f\times(g)^{-1}\) and taking the derivative of the latter.
  10. The chain rule states that for the first derivative of the composite of two functions, \(\bigl(f\bigl(g(x)\bigr)\bigr)' = f'\bigl(g(x)\bigr)g'(x)\,.\) But what about the second derivative of the composite of two functions? Determine a formula for \(\bigl(f\bigl(g(x)\bigr)\bigr)''\,.\)
  11. Typically trig functions expect their argument in radian measure. Define \(\operatorname{degsin}\) to be sine function that expects its argument in degree measure. I.e. for \(f(x) = \frac{\pi}{180}x\) we have \(\operatorname{degsin} = \sin \circ f\,.\) What is the derivative of \(\operatorname{degsin}\,?\)

  12. Using the fact that \(\lim_{x \to 0} \sin(x)/x = 1\) demonstrate how to algebraically determine the values of the following limits.

    \(\displaystyle \lim\limits_{x \to 0} \frac{\sin(3x)}{7x} \)
    \(\displaystyle \lim\limits_{x \to 0} \frac{\sin(3x)}{\sin(7x)} \)
    \(\displaystyle \lim\limits_{x \to 0} \frac{x\cos(x)}{\tan(x)} \)
    \(\displaystyle \lim\limits_{x \to 0} \frac{\sin^2(2x)}{x^2} \)
  13. So far we’ve been taking the fact that the derivative of sine is cosine as something we just know. But this fact is not manifest. It follows from the definition of the derivative as a limit. Use the definition of the derivative as a limit to prove that the derivative of sine is cosine. Hints: you’ll likely want to use these two facts in your proof: \[ \sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b) \qquad\text{and}\qquad \lim_{x \to 0} \frac{\cos(x)-1}{x} = 0 \] The first of these is the sum-of-angles formula for sine (do you remember how to prove that?), and the second follows from \(\lim_{x \to 0}\frac{\sin(x)}{x} = 1.\)
  14. James Stewart Without appealing to technology, evaluate the limit \[ \lim\limits_{x \to 0} \frac{\sin\big((3+x)^2\big) - \sin(9)}{x}\,. \]
  15. James Stewart Determine the domain of \(f\), and calculate a formula for \(f'(x),\) for \(f\) defined as \[ f(x) = \sqrt{1- \sqrt{2- \sqrt{3-x } } }\,. \]

  16. We denote the \(n^\text{th}\) derivative of a function \(f\) as \(f^{(n)}.\) For example, \(f^{(13)}(x)\) denotes the thirteenth derivative of \(f,\) and denotes it better than \(f'''''''''''''(x).\)

    What is the first derivative of the function \(f(x) = \tfrac{1}{1-x}\,?\) What is the fourth derivative of this function? What is a formula for \(f^{(n)}(x)?\)

  17. What’s a formula for the \(n^{\text{th}}\) derivative of \(x\sin(x)\)?
  18. What’s a formula for the \(n^\text{th}\) derivative of tangent? What about secant?
  19. James Stewart What’s a formula for the \(n^\text{th}\) derivative of \(f(x) = x^n/(1-x)\,?\)