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Calculus 1

  1. Find \(\frac{dy}{dx}\) when \(x^3 + 3y^4 = 2x + 7\)

  2. Find the derivative of \(y\) with respect to \(x\) for the following equation
    \(y(x+4) = x^2 - 3\)

  3. Find \(\frac{dy}{dx}\) and the slope of the tangent line at \((-2, 1)\) for the curve given by
    \(2x^2 - 3y^3 = 5\)

  4. Determine the first and second derivatives, \(\frac{dy}{dx}\) and \(\frac{d^{2}y}{dx^2}\) for the following equation
    \(x^2 + xy = 4\)

  5. Find \(\frac{dy}{dx}\) for \(x^2 + y^3 = \log{(x + y)}\)

  6. Find \(\frac{dy}{dx}\) and the slope of the tangent line at (0,3) for the curve given by
    \(y^3 + x^{2}y^{5} - x^4 = 27\)

  7. Find the tangent line to the curve \(xy + \ln{(xy^2)} = 1\) at the point \((1,1)\)

  8. For the following equation, differentiate implicitly to find \(\frac{dy}{dx}\)
    \(e^{(x + y)} = \sin{(x)} + \cos{(y)}\)

  9. Show that \(\frac{d}{dx}(\arcsin{x}) = \frac{x^{\prime}}{\sqrt{1 - x^2}}\)

  10. Show that for \(y = \cos^{-1}(x)\) the first derivative, \(\frac{dy}{dx} = \frac{1}{x^2 + 1}\)

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