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Home / Calculus I / Derivatives / Product and Quotient Rule
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Section 3-4 : Product and Quotient Rule

For problems 1 – 6 use the Product Rule or the Quotient Rule to find the derivative of the given function.

  1. \(f\left( t \right) = \left( {4{t^2} - t} \right)\left( {{t^3} - 8{t^2} + 12} \right)\) Solution
  2. \(y = \left( {1 + \sqrt {{x^3}} } \right)\,\left( {{x^{ - 3}} - 2\sqrt[3]{x}} \right)\) Solution
  3. \(h\left( z \right) = \left( {1 + 2z + 3{z^2}} \right)\left( {5z + 8{z^2} - {z^3}} \right)\) Solution
  4. \(\displaystyle g\left( x \right) = \frac{{6{x^2}}}{{2 - x}}\) Solution
  5. \(\displaystyle R\left( w \right) = \frac{{3w + {w^4}}}{{2{w^2} + 1}}\) Solution
  6. \(\displaystyle f\left( x \right) = \frac{{\sqrt x + 2x}}{{7x - 4{x^2}}}\) Solution
  7. If\(f\left( 2 \right) = - 8\), \(f'\left( 2 \right) = 3\), \(g\left( 2 \right) = 17\) and \(g'\left( 2 \right) = - 4\) determine the value of \({\left( {f\,g} \right)^\prime }\left( 2 \right)\). Solution
  8. If \(f\left( x \right) = {x^3}g\left( x \right)\), \(g\left( { - 7} \right) = 2\), \(g'\left( { - 7} \right) = - 9\) determine the value of \(f'\left( { - 7} \right)\). Solution
  9. Find the equation of the tangent line to \(f\left( x \right) = \left( {1 + 12\sqrt x } \right)\left( {4 - {x^2}} \right)\) at \(x = 9\). Solution
  10. Determine where \(\displaystyle f\left( x \right) = \frac{{x - {x^2}}}{{1 + 8{x^2}}}\) is increasing and decreasing. Solution
  11. Determine where \(V\left( t \right) = \left( {4 - {t^2}} \right)\left( {1 + 5{t^2}} \right)\) is increasing and decreasing. Solution