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Calc 1 Concepts: Derivatives
● The derivative is a fundamental concept in Calculus that describes the instantaneous
rate of change of a function at a certain point
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e
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● Geometrically, the derivative of a function f(x) at a point x = a is equal to the slope of
the tangent line to the curve at that point
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● There are several techniques for finding derivatives, including the power rule,
product rule, quotient rule, and chain rule
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● (Include a diagram of a graph of a function with a polynomial curve to illustrate the
concept of the power rule)
● The product rule states that if f(x) = g(x)h(x), then f'(x) = g'(x)h(x) + g(x)h'(x)
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● The chain rule is used when a function is composed of multiple functions nested
within each other
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● Overall, the derivative is a powerful tool in Calculus that allows us to understand the
behavior of functions in a variety of applications
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○ The first derivative test can be used to find relative extrema (i
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, local
maxima and minima) of a function
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● The derivative can also be used to find the rate of change of a function with respect
to time, which is useful in a variety of real-world applications
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e
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● The second derivative can be used to find the inflection points of a function (i
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,
points where the concavity of the function changes)
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● Partial derivatives describe the rate of change of a function with respect to one of its
variables, while holding all other variables constant
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● The gradient is a vector-valued function that describes the direction of maximum
increase of a function at a certain point
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● Overall, derivatives are a critical concept in Calculus that are used to analyze the
behavior of functions in a variety of applications
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