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### Section 3-3 : Differentiation Formulas

For problems 1 – 12 find the derivative of the given function.

1. $$f\left( x \right) = 6{x^3} - 9x + 4$$ Solution
2. $$y = 2{t^4} - 10{t^2} + 13t$$ Solution
3. $$g\left( z \right) = 4{z^7} - 3{z^{ - 7}} + 9z$$ Solution
4. $$h\left( y \right) = {y^{ - 4}} - 9{y^{ - 3}} + 8{y^{ - 2}} + 12$$ Solution
5. $$y = \sqrt x + 8\,\sqrt{x} - 2\,\sqrt{x}$$ Solution
6. $$f\left( x \right) = 10\,\sqrt{{{x^3}}} - \sqrt {{x^7}} + 6\,\sqrt{{{x^8}}} - 3$$ Solution
7. $$\displaystyle f\left( t \right) = \frac{4}{t} - \frac{1}{{6{t^3}}} + \frac{8}{{{t^5}}}$$ Solution
8. $$\displaystyle R\left( z \right) = \frac{6}{{\sqrt {{z^3}} }} + \frac{1}{{8{z^4}}} - \frac{1}{{3{z^{10}}}}$$ Solution
9. $$z = x\left( {3{x^2} - 9} \right)$$ Solution
10. $$g\left( y \right) = \left( {y - 4} \right)\left( {2y + {y^2}} \right)$$ Solution
11. $$\displaystyle h\left( x \right) = \frac{{4{x^3} - 7x + 8}}{x}$$ Solution
12. $$\displaystyle f\left( y \right) = \frac{{{y^5} - 5{y^3} + 2y}}{{{y^3}}}$$ Solution
13. Determine where, if anywhere, the function $$f\left( x \right) = {x^3} + 9{x^2} - 48x + 2$$ is not changing. Solution
14. Determine where, if anywhere, the function $$y = 2{z^4} - {z^3} - 3{z^2}$$ is not changing. Solution
15. Find the tangent line to $$\displaystyle g\left( x \right) = \frac{{16}}{x} - 4\sqrt x$$ at $$x = 4$$. Solution
16. Find the tangent line to $$f\left( x \right) = 7{x^4} + 8{x^{ - 6}} + 2x$$ at $$x = - 1$$. Solution
17. The position of an object at any time t is given by $$s\left( t \right) = 3{t^4} - 40{t^3} + 126{t^2} - 9$$.
1. Determine the velocity of the object at any time t.
2. Does the object ever stop changing?
3. When is the object moving to the right and when is the object moving to the left?
Solution
18. Determine where the function $$h\left( z \right) = 6 + 40{z^3} - 5{z^4} - 4{z^5}$$ is increasing and decreasing. Solution
19. Determine where the function $$R\left( x \right) = \left( {x + 1} \right){\left( {x - 2} \right)^2}$$ is increasing and decreasing. Solution
20. Determine where, if anywhere, the tangent line to $$f\left( x \right) = {x^3} - 5{x^2} + x$$ is parallel to the line $$y = 4x + 23$$. Solution