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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. $$y = \left( {1 + \sqrt {{x^3}} } \right)\,\left( {{x^{ - 3}} - 2\sqrt{x}} \right)$$ 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