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 study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
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