The derivative of a function is itself a function, so we can find the derivative of a derivative. For example, the derivative of a position function is the rate of change of position, or velocity. The derivative of velocity is the rate of change of velocity, which is acceleration. The new function obtained by differentiating the derivative is called the second derivative. Furthermore, we can continue to take derivatives to obtain the third derivative, fourth derivative, and so on. Collectively, these are referred to as
higher-order derivatives . The notation for the higher-order derivatives of
can be expressed in any of the following forms:
It is interesting to note that the notation for
may be viewed as an attempt to express
more compactly. Analogously,
The position of a particle along a coordinate axis at time
(in seconds) is given by
(in meters). Find the function that describes its acceleration at time
The derivative of a function
is the function whose value at
is
The graph of a derivative of a function
is related to the graph of
Where
has a tangent line with positive slope,
Where
has a tangent line with negative slope,
Where
has a horizontal tangent line,
If a function is differentiable at a point, then it is continuous at that point. A function is not differentiable at a point if it is not continuous at the point, if it has a vertical tangent line at the point, or if the graph has a sharp corner or cusp.
Higher-order derivatives are derivatives of derivatives, from the second derivative to the
derivative.
Key equations
The derivative function
For the following exercises, use the definition of a derivative to find
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include:
1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body
the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature