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First Derivative: $ f’(x) $ or $ \frac{df}{dx} $
The first derivative, $ f’(x) $ or $ \frac{df}{dx} $ , tells us the instantaneous slope of the tangent line to the graph of $ f(x) $ at a given point.
Second Derivative: $ f’’(x) $ or $ \frac{d^2f}{dx^2} $
The second derivative, $ f’’(x) $ or $ \frac{d^2f}{dx^2} $ , is the derivative of the first derivative. It tells us the rate of change of the slope of the tangent line.
- Concavity: The sign of the second derivative helps determine the concavity of a function:
- $ f’’(x) > 0 $ : The graph of $ f(x) $ is concave up (shaped like a cup).
- $ f’’(x) < 0 $ : The graph of $ f(x) $ is concave down (shaped like a frown).
- Inflection Points: Points where the concavity changes are called inflection points. These occur where $ f’’(x) = 0 $ or is undefined.
Third Derivative and Beyond: $ f’’’(x) $ or $ \frac{d^3f}{dx^3} $
The third derivative, $ f’’’(x) $ or $ \frac{d^3f}{dx^3} $ , is the derivative of the second derivative. It tells us the rate of change of the concavity.
We can continue taking derivatives to get higher-order derivatives: $ f^{(4)}(x) $ , $ f^{(5)}(x) $ , and so on.
Higher-order derivatives are often used in physics and engineering to describe the motion of objects. For example, the second derivative of position with respect to time is acceleration.
Example
Let’s consider the function $ f(x) = x^3 - 3x^2 + 2x $ .
- First Derivative: $ f’(x) = 3x^2 - 6x + 2 $
- Second Derivative: $ f’’(x) = 6x - 6 $
- Third Derivative: $ f’’’(x) = 6 $
We can see that the second derivative is zero at $ x = 1 $ . This means that there is an inflection point at $ x = 1 $ . Furthermore, the second derivative is positive for $ x > 1 $ and negative for $ x < 1 $ , indicating that the function is concave up for $ x > 1 $ and concave down for $ x < 1 $ .
Applications
Higher-order derivatives have various applications in different fields:
- Physics: They describe the motion of objects (acceleration, jerk, etc.).
- Engineering: They are used in optimization problems and modeling physical systems.
- Economics: They are used in marginal analysis.
- Statistics: They are used in curve fitting and data analysis.
Summary
Higher-order derivatives provide valuable information about the behavior of a function beyond its rate of change. They help us understand concavity, inflection points, and other important properties that are crucial in various applications.