Derivatives of Polynomials

When calculating the derivatives of polynomials (especially more complex ones), it might get a little tedious using First Principles to determine every one. As a result, there are a set of rules used to make this process more straightforward:

Rule Original Derivative
Constant \(f(x) = c\) \(f'(x) = 0\)
Power \(f(x) = x^n\) \(f'(x) = nx^{n-1}\)
Constant Multiple \(f(x) = K\cdot g(x)\) \(f'(x) = K\cdot g'(x)\)
Sum \(f(x) = g(x) + H(x)\) \(f'(x) = g'(x) + H'(x)\)
Difference \(f(x) = g(x) - H(x)\) \(f'(x) = g'(x) - H'(x)\)


Make sure to use the calculator here in order to verify your answers for the Power Rule!


Example

Differentiate the following functions. Simplify first if necessary:

  1. \(y = 5x^6 -4x^3 + 6\)
  2. \(f(x) = -3x^5 + 8\sqrt{x} - \cfrac{9}{x}\)

i. We can take the derivative of each individual term using the Sum Rule. We can also differentiate the first 2 terms (\(5x^6\) and \(-4x^3\)) using the Power Rule. Additionally, we can differentiate the last term using the Constant Rule:

\(y' = \textcolor{red}{3}(4x^{\textcolor{red}{3}-1}) + 0\)

\(y' = 30x^5 -12x^2\)

Therefore, we can determine that \(y' = 30x^5 -12x^2\).


ii. First, we can rewrite the function as such:

\(f(x) = -3x^5 + 8x^{\frac{1}{2}} - 9x^{-1}\)

Next, we can take the derivative of each individual term using the Sum/Difference Rules. We can also differentiate each term by using Power Rule:

\(f'(x) = \textcolor{red}{5}(-3x^{\textcolor{red}{5}-1}) + \cfrac{\textcolor{red}{1}}{\textcolor{red}{2}}(8x^{\textcolor{red}{\frac{1}{2}}-1}) - (\textcolor{red}{-1})(9x^{\textcolor{red}{-1}-1})\)

\(f'(x) = -15x^4 + 4x^{-\frac{1}{2}} + 9x^{-2}\)

Finally, we can express the derivative in simplest form by rewriting it with positive exponents:

\(f'(x) = -15x^4 + \cfrac{4}{\sqrt{x}} + \cfrac{9}{x^2}\)

Therefore, we can determine that \(f'(x) = -15x^4 + \cfrac{4}{\sqrt{x}} + \cfrac{9}{x^2}\).


Differentiate each function. Simplify first if necessary.

\(g(x) = (2x-3)(x+1)\)


\(h(x) = \cfrac{-8x^6+8x^2}{4x^5}\)


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