Exercises - The Chain Rule

  1. Differentiate:

    1. $\displaystyle{f\,(x) = (x^3 - x + 1)^{10}}$

    2. $\displaystyle{g(x) = \sqrt{25-x^2}}$

    3. $\displaystyle{h(x) = \tan^2 x}$

    4. $\displaystyle{f(x) = \cos(4x^2)}$

    5. $\displaystyle{g(x) = e^{4x+3}}$

    6. $\displaystyle{h(x) = \sin(\cos \sqrt{x})}$

    7. $\displaystyle{f(x) = \ln \sqrt{x+1}}$

    8. $\displaystyle{g(x) = \ln \left[ \frac{e^x (x^2-4)}{\sqrt{x}}\right]}$

    9. $\displaystyle{y = \textrm{Arctan } \sqrt{x}}$

    10. $\displaystyle{y = \textrm{Arcsin } \left( \frac{1}{x} \right)}$

  2. Given $f\,(x) = (4-x)^{4/3}$
    1. Find the equation of the normal line to the graph of $y=f\,(x)$ at $x=-4$
    2. At what point(s) is the tangent line to the graph of $y=f\,(x)$ horizontal?