Introduction to Derivatives | Chapter 1.5 | Calculus and Vectors Course

Hello everybody! Today we will finally explore the basics of differentiation and derivatives. We will look at the first principles definition of the derivative and do some examples using it. We will also look at which types of graphs might not have a derivative!

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Chapter 1.5 Slideshow:

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differentiation
Differentiation is one of the most fundamental and powerful operations of calculus
The output of this operation is called the derivative
The derivative of a function can be found using the first principles definition of the derivative
f’(x) = lim [f(x+h) - f(x)]/h

When this limit is simplified by letting h→ 0, the resulting expression is expressed in terms of x. You can use this expression to determine the derivative of the function at any x-value that is in the function’s domain
You might’ve noticed the new notation f’(x), well this is referred to Lagrange notation. Another notation you need to know is dy/dx, which is referred as the Leibniz Notation.
Leibniz Notation can also be used to evaluate the value of the derivative when x = a, and it looks like this:
dy
dx x = a

Leibniz Notation can also be used to showcase a relationship between different things for
example the volume of a gas, V, and temperature, t, your notation would be
dV/dt, or volume with respect to temperature.

Differentiation Contd
A derivative may not exist at every point on a curve. For example, discontinuous functions are non-differentiable at the point(s) where they are discontinuous.
For example, at any VA’s or holes
There are also continuous functions that may not be differentiable at some points
For example, sometimes the right hand side limit will vary from the left-hand side limit, and thus the derivative does not approach the same value, and thus his function is non-differentiable at P even though the function is continuous.
These points are represented by cusps or corners on graphs!

Derivatives Examples

state the derivative of f(x) = 1/x
Find dy/dx of 1/2x3
Differentiate y = -4/3x
Evaluate f’(-6) if f(x) = x3

Key Concepts
The derivative of a function can be found using the first principles definition of the derivative f’(x) = lim [f(x+h) - f(x)]/h
Lagrange Notation: f’(x)
Leibniz Notation: dy/dx
If the derivative does not exist at a point on the curve, the function is non-differentiable at that x-value. This can occur at points where the function is discontinuous or in cases where the function has an abrupt change, which is represented by a cusp or corner on a graph.

Homework
Page 58: 1, 4, 5, 7, 8, 14ace, 16
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