Section 3.3 : Differentiation Formulas

For problems 1 – 20 find the derivative of the given function.

1. \(g\left( x \right) = 8 - 4{x^3} + 2{x^8}\)
2. \(f\left( z \right) = {z^{10}} - 7{z^5} + 2{z^3} - {z^2}\)
3. \(y = 8{x^4} - 10{x^3} - 9x + 4\)
4. \(f\left( x \right) = 3{x^{ - 4}} + {x^4} - 3x\)
5. \(R\left( t \right) = 9{t^{10}} + 8{t^{ - 10}} + 12\)
6. \(h\left( y \right) = 3{y^{ - 6}} - 8{y^{ - 3}} + 9{y^{ - 1}}\)
7. \(g\left( t \right) = {t^{ - 7}} + 2{t^{ - 3}} - 6{t^{ - 2}} + 8{t^4} - 1\)
8. \(z = \sqrt[6]{x} - 7\,\sqrt[4]{x} + 3\sqrt x \)
9. \(f\left( x \right) = 7\,\sqrt[9]{{{x^4}}} - 2\,\sqrt[2]{{{x^7}}} + \sqrt[3]{{{x^4}}}\)
10. \(\displaystyle h\left( y \right) = 6\sqrt y + \sqrt[6]{{{y^5}}} + \frac{7}{{\sqrt[9]{{{y^2}}}}}\)
11. \(\displaystyle g\left( z \right) = \frac{4}{{{z^2}}} + \frac{1}{{7{z^5}}} - \frac{1}{{2z}}\)
12. \(\displaystyle y = \frac{2}{{3{t^9}}} + \frac{1}{{7{t^3}}} - 9{t^2} - \sqrt {{t^3}} \)
13. \(\displaystyle W\left( x \right) = {x^3} - \frac{1}{{{x^6}}} + \frac{1}{{\sqrt[5]{{{x^2}}}}}\)
14. \(g\left( w \right) = \left( {w - 5} \right)\left( {{w^2} + 1} \right)\)
15. \(h\left( x \right) = \sqrt x \left( {1 - 9{x^3}} \right)\)
16. \(f\left( t \right) = {\left( {3 - 2{t^3}} \right)^2}\)
17. \(g\left( x \right) = \left( {1 + 2x} \right)\left( {2 - x + {x^2}} \right)\)
18. \(\displaystyle y = \frac{{4 - 8x + 2{x^2}}}{x}\)
19. \(\displaystyle Y\left( t \right) = \frac{{{t^4} - 2{t^2} + 7t}}{{{t^3}}}\)
20. \(\displaystyle S\left( w \right) = \frac{{{w^2}\left( {2 - w} \right) + {w^5}}}{{3w}}\)

For problems 21 – 26 determine where, if anywhere, the function is not changing.

1. \(f\left( x \right) = 2{x^3} - 9{x^2} - 108x + 14\)
2. \(u\left( t \right) = 45 + 300{t^2} + 20{t^3} - 3{t^4}\)
3. \(Q\left( t \right) = {t^3} - 9{t^2} + t - 10\)
4. \(h\left( w \right) = 2{w^3} + 3{w^2} + 4w + 5\)
5. \(g\left( x \right) = 9 + 8{x^2} + 3{x^3} - {x^4}\)
6. \(G\left( z \right) = {z^2}{\left( {z - 1} \right)^2}\)
7. Find the tangent line to \(f\left( x \right) = 3{x^5} - 4{x^2} + 9x - 12\) at \(x = - 1\).
8. Find the tangent line to \(\displaystyle g\left( x \right) = \frac{{{x^2} + 1}}{x}\) at \(x = 2\).
9. Find the tangent line to \(h\left( x \right) = 2\sqrt x - 8\,\sqrt[4]{x}\) at \(x = 16\).
10. The position of an object at any time t is given by \(s\left( t \right) = 3{t^4} - 44{t^3} + 108{t^2} + 20\).
    1. Determine the velocity of the object at any time t.
    2. Does the position of the object ever stop changing?
    3. When is the object moving to the right and when is the object moving to the left?
11. The position of an object at any time t is given by \(s\left( t \right) = 1 - 150{t^3} + 45{t^4} - 2{t^5}\).
    1. Determine the velocity of the object at any time t.
    2. Does the position of the object ever stop changing?
    3. When is the object moving to the right and when is the object moving to the left?
12. Determine where the function \(f\left( x \right) = 4{x^3} - 18{x^2} - 336x + 27\) is increasing and decreasing.
13. Determine where the function \(g\left( w \right) = {w^4} + 2{w^3} - 15{w^2} - 9\) is increasing and decreasing.
14. Determine where the function \(V\left( t \right) = {t^3} - 24{t^2} + 192t - 50\) is increasing and decreasing.
15. Determine the percentage of the interval \(\left[ { - 6,4} \right]\) on which \(f\left( x \right) = 7 + 10{x^3} - 5{x^4} - 2{x^5}\) is increasing.
16. Determine the percentage of the interval \(\left[ { - 5,2} \right]\) on which \(f\left( x \right) = 3{x^4} - 8{x^3} - 144{x^2}\) is decreasing.
17. Is \(h\left( x \right) = 3 - x + {x^2} + 2{x^3}\) increasing or decreasing more on the interval \(\left[ { - 1,1} \right]\)?
18. Determine where, if anywhere, the tangent line to \(f\left( x \right) = 12{x^2} - 9x + 3\) is parallel to the line \(y = 1 - 7x\).
19. Determine where, if anywhere, the tangent line to \(f\left( x \right) = 8 + 4x + {x^2} - 2{x^3}\) is perpendicular to the line \(\displaystyle y = - \frac{1}{4}x + \frac{8}{3}\).
20. Determine where, if anywhere, the tangent line to \(f\left( x \right) = \sqrt[3]{x} - 8x\) is perpendicular to the line \(y = 2x - 11\).
21. Determine where, if anywhere, the tangent line to \(\displaystyle f\left( x \right) = \frac{{13x}}{9} + \frac{1}{x}\) is parallel to the line \(y = x\).