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Wize University Calculus 2 Textbook > Review: Derivatives

Basics of Derivatives

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Interpretation of Derivatives

The derivative of a function y=f(x)y=f\left(x\right) is represented by y′, f′(x), or dydxy',\ f'\left(x\right),\ \text{or}\ \frac{dy}{dx}.
  1. Graphically: The derivative of a function f(x)f\left(x\right) at a point aa is the slope of the tangent to that curve at aa
  2. Real-world application: It represents the rate of change of that function at that point
Note:
The curve has a
horizontal tangent
at a point ⇔\Leftrightarrow the derivative at that point is
0

Limit/Formal Definitions

The derivative of a function f(x)f\left(x\right): f′(x)=lim⁡h→0 f(x+h)−f(x)h\displaystyle f'\left(x\right)=\lim_{h\to0}\ \frac{f\left(x+h\right)-f\left(x\right)}{h}

The derivative of a function f(x)f\left(x\right) at a point aa: f′(a)=lim⁡x→a f(x)−f(a)x−a\displaystyle f'\left(a\right)=\lim_{x\to a}\ \frac{f\left(x\right)-f\left(a\right)}{x-a}

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Differentiability

If this limit exists at a point aa, we say that f(x)f\left(x\right) is differentiable at aa.
If this limit exists for all points xx, we say that f(x)f(x) is a differentiable function.

Not Differentiable

  • If it is not continuous
  • If it has a corner
  • If it has a vertical tangent

Watch Out!
If a function is differentiable at a point, then it is continuous at that point.
But this is NOT necessarily true the other way around.

Using the formal definition of a derivative, the derivative of f(x)=x2f\left(x\right)=\frac{\sqrt{x}}{2} simplifies to

Practice: Differentiability

Which of the following is/are NOT differentiable at x=0x=0?

(Select all that apply)