Derivative as a Function

Reading

• Sections 3.2

Practice problems

• Section 3.2: 5, 7, 17, 19, 21, 35, 51, 53
• To turn in: 3.2 16, 20, 26, 36, 66
• In class: 3.2 43, 45

Notes

The derivative as a function

• If the derivative \(f'(a)\) exists for all \(a\) in some interval, then we can look at the function \(f'(x)\) and study its properties.
• We often do this by computing \(\displaystyle \lim_{h\to 0}\frac{f(x+h) - f(x)}{h}\).
• Example: Derivative of \(f(x) = 2x^2 + x\) is \(4x+1\).
• Example 2: Derivative of \(f(x) = \frac{1}{x^2}\) is \(-\frac{2}{x^3}\).
• Alternative notation: \(\frac{df}{dx}\), \(\frac{dy}{dx}\), \(\frac{d}{dx}f\), \(\left.\frac{df}{dx}\right|_{x=4}\).

• Power Rule:

  \[\frac{d}{dx} x^n = x^{n-1}\]

• Prove the power rule using both derivative definitions.

• Linearity rules:

  \[(f+g)' = f' + g'\] \[(kf)' = k f'\]

• Comparing the graph of the derivative with that of the function: The value of the derivative at a point relates to the slope of the tangent of the function at that point. Look for whether the slope increases/decreases.
• When is the tangent line horizontal?

• Differentiability requires continuity:

  If \(f\) is differentiable at \(x=a\), then \(f\) is also continuous at \(a\).

  Alternatively, if \(f\) is not continuous at \(a\) then it cannot possibly be differentiable there.

• There are functions that are continuous but not differentiable (e.g. \(|x|\)).