Definition

Recall that the derivative of a function \(f(x)\) at \(x=a\) is denoted by \(f'(a)\), \[f'(a) = \lim_{h\to 0}\dfrac{f(a+h) - f(a)}{(a + h) - a}= \lim_{h\to 0}\dfrac{f(a+h) - f(a)}{ h}.\]

The derivative of a function \(f(x)\) at \(x=a\) is denoted by \(f'(a)\), \[f'(x) = \lim_{h\to 0}\dfrac{f(x+h) - f(x)}{(x+h) - x}=\lim_{h\to 0}\dfrac{f(x+h) - f(x)}{h}. \] The derivative of \(y = f(x)\) is also commonly denoted as \(\dfrac{dy}{dx}\).

Example 1. Let \(f(x) = 2\). Find \(f'(x)\).



Example 2. Let \(f(x) = x^2\). Find \(f'(x)\).



Example 3. Let \(f(x) = x^3\). Find \(f'(x)\).



Example 4. Let \(f(x) = x^4\). Find \(f'(x)\).



Can you tell which one is \(f(x)\) and which one is \(f'(x)\)?

Basic Rules

  • Power rule: \((x^n)' = nx^{n-1}\).
  • Sum rule: \(\big(f(x) \pm g(x)\big)' = f'(x) \pm g'(x)\).
  • Constant multiple: \(\big(cf(x)\big)' = cf'(x)\).

Differentiability

If the derivatives exist everywhere in the domain of \(f(x)\), then we say \(f(x)\) is differentiable.

Intuitively, differentiability suggests the function is not only continuous but also “smooth”.

Is \(f(x)=|x|\) differentiable at \(x=0\)?

Higher-order Derivatives

Let \(y=f(x)\), the higher order derivatives of \(f(x)\) are recursively defined as

\[f''(x) = \big(f'(x)\big)'\]

\[f'''(x) = \big(f''(x)\big)'\]

\[f^{(4)}(x) = \big(f'''(x)\big)'\]

\[\vdots\]

Exponential Function

Example 5 Let \(f(x) = e^x\). Find \(f'(x)\).



Practice Problems

  1. Find the derivative of the following functions by the definition of derivative.

    • \(f(x)=x^2 - 2x^3\)
    • \(f(x)=\dfrac{1 - 2x}{3 + x}\)

  2. Find the derivative of the following functions by the derivative rules.

    • \(f(x)=x^{5/3} - x^{2/3}\)
    • \(f(x)=3e^x+x^e\)
    • \(f(x)=e^{x+2} + 1\)
    • \(f(x)=\sqrt x - \frac 1 x + \frac{1}{x^2}\)
    • \(f(x)=\dfrac{x^2 + 4x + 3}{\sqrt x}\)
    • \(f(x)=\pi^2\)

  3. Identify \(f(x)\) and \(f'(x)\) on the following graphs.



  4. Identify \(f(x), f'(x)\), and \(f''(x)\) on the following graphs.