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Velocity, Acceleration and Second Order Derivatives

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Velocity, Acceleration and Second Order Derivatives

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The second-order derivative: differentiating the first-order derivative of the function
The slope of the slope
EXAMPLE

The relationship between velocity, acceleration, and displacement

NOTICE!!!
Speed is a scalar quantity. It describes the magnitude of motion but does not describe the direction.
Velocity has both magnitude and direction.

QUESTION ⑤
Given the graph of 𝑠(𝑡), figure out where 𝑣(𝑡) and 𝑎(𝑡) are + or –, and use this information to state when the particle is speeding up and slowing down.

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If f(x) and g(x) are both differentiable functions and g(x) does not equal zero, then the derivative of their quotient also exists

EXAMPLE

QUESTION ④

h(x) is defined as f(x)/g(x), please find h'(4).

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The Composition of Functions
"Function Composition" is applying one function to the results of another

The result of g(x) is sent through f(x), so the f(x) is an outer function, and the g(x) is an inner function

Chain Rule: Differentiate outer function with respect to the inner function, and then multiply by the derivative of the inner function.

EXAMPLE
Given two differentiable functions 𝑔(𝑥) and f(𝑥), the derivative of the composite function h(x)=f(g(x)) is h’(x)=f’(g(x))*g’(x)
ALSO CALLED: Lagrange Notation
Restriction:
1.     Function g should be differentiable at x

2.     Function f should be differentiable at f(x)

Leibniz Notation
Suppose y is a function of x: y = g(x)
z is a function of y: z = f(y)
Then z is a function of x: z = f(y) = f(g(x))

QUESTION ③
Please differentiate the function h(x)=(5-6x)⁵

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The product rule:
if f(x) and g(x) are both differentiable functions, then the derivative of their product also exists.

Form

Remember it through a sentence: product rule is the derivative of the first times the second plus derivative of the second times the first.

Alternative notation

EXAMPLE
Please differentiate the function f(x)

The product of three functions

Generalized Power Rule

EXAMPLE

QUESTION ②
Please find h'(-1) where h(𝑥) = (5𝑥³ + 7𝑥² + 3)(2𝑥² + 𝑥 + 6)

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The Power rule tells us how to differentiate expressions of the form xn
n, x ∈ R

Example: What is the derivative of x2
For x² we use the Power Rule with n=2:
The derivative of   x2 = 2x(2−1)
= 2x¹
= 2x
Answer: the derivative of x² is 2x

Multiply by power, then reduce power by 1

Constant Function Rule

The derivative of any constant function is always 0

Three basic differential rules

QUESTION ①：
H(x) is defined as 2f(x)-3g(x)+5, how can you derivative the H(x)? And if f'(3)=1, g'(3)=5, what is the value of H'(3)?

How to find the equation of the tangent line to a point on the graph of f(x)?
1. Find derivative function f’(x)
2. Find the slope of the tangent line using: m=f’(x₀)
3. Use the slope-point formula to get the equation of the tangent line: y-f(x₀)=m(x-x₀)
The geometric meaning of the first-order derivative is the slope of the tangent line.

EXAMPLE: Find the equation of the tangent line to the curve
at the point (1,3)

Normal Line
The normal line is the line that perpendicular to the tangent line at the point of tangency

If mT is the slope of the tangent line, then the slope of the normal line mN

Piece-wise defined functions

A discontinuous function must be non-differentiable, and a continuous function must have the equal slope of the tangent line for the left branch and the right branch to be differentiable

EXAMPLE