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Section 3.3 : Differentiation Formulas
For problems 1 – 12 find the derivative of the given function.
- \(f\left( x \right) = 6{x^3} - 9x + 4\) Solution
- \(y = 2{t^4} - 10{t^2} + 13t\) Solution
- \(g\left( z \right) = 4{z^7} - 3{z^{ - 7}} + 9z\) Solution
- \(h\left( y \right) = {y^{ - 4}} - 9{y^{ - 3}} + 8{y^{ - 2}} + 12\) Solution
- \(y = \sqrt x + 8\,\sqrt[3]{x} - 2\,\sqrt[4]{x}\) Solution
- \(f\left( x \right) = 10\,\sqrt[5]{{{x^3}}} - \sqrt {{x^7}} + 6\,\sqrt[3]{{{x^8}}} - 3\) Solution
- \(\displaystyle f\left( t \right) = \frac{4}{t} - \frac{1}{{6{t^3}}} + \frac{8}{{{t^5}}}\) Solution
- \(\displaystyle R\left( z \right) = \frac{6}{{\sqrt {{z^3}} }} + \frac{1}{{8{z^4}}} - \frac{1}{{3{z^{10}}}}\) Solution
- \(z = x\left( {3{x^2} - 9} \right)\) Solution
- \(g\left( y \right) = \left( {y - 4} \right)\left( {2y + {y^2}} \right)\) Solution
- \(\displaystyle h\left( x \right) = \frac{{4{x^3} - 7x + 8}}{x}\) Solution
- \(\displaystyle f\left( y \right) = \frac{{{y^5} - 5{y^3} + 2y}}{{{y^3}}}\) Solution
- Determine where, if anywhere, the function \(f\left( x \right) = {x^3} + 9{x^2} - 48x + 2\) is not changing. Solution
- Determine where, if anywhere, the function \(y = 2{z^4} - {z^3} - 3{z^2}\) is not changing. Solution
- Find the tangent line to \(\displaystyle g\left( x \right) = \frac{{16}}{x} - 4\sqrt x \) at \(x = 4\). Solution
- Find the tangent line to \(f\left( x \right) = 7{x^4} + 8{x^{ - 6}} + 2x\) at \(x = - 1\). Solution
- The position of an object at any time t is given by \(s\left( t \right) = 3{t^4} - 40{t^3} + 126{t^2} - 9\).
- Determine the velocity of the object at any time t.
- Does the object ever stop changing?
- When is the object moving to the right and when is the object moving to the left?
- Determine where the function \(h\left( z \right) = 6 + 40{z^3} - 5{z^4} - 4{z^5}\) is increasing and decreasing. Solution
- Determine where the function \(R\left( x \right) = \left( {x + 1} \right){\left( {x - 2} \right)^2}\) is increasing and decreasing. Solution
- 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