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Practice Problems for 3/4

  1. Find \(\frac{dy}{dx}\) for \(y=e^{\sec x}\).

  2. Each of the following functions can be written as \((f\circ g)(x)\). Define an appropriate \(f\) and \(g\) for each function.
    1. \(5^{-1/x}\)
    2. \(\cot(x^3)\)
    3. \(\left( \frac{x-6}{x^7+9} \right)^4\)

  3. Let \(y=\frac{x^2}{\sin x}\).
    1. Calculate \(\frac{dy}{dx}\) using the quotient rule.
    2. Calculate \(\frac{dy}{dx}\) using the product rule by writing \(y=x^2(\sin x)^{-1}\).

  4. Let \(y=m(x)=\cos(\sqrt{x^4+x})\).

    1. \(m(x)\) is the composition of three functions. Find \(f,\;g,\;\) and \(h\) such that \(m(x)=(h\circ f\circ g)(x)\).
    2. Find \(\frac{dy}{dx}\).