first derivative
- Noun:
- The result of mathematical differentiation: The "first derivative" is the fundamental outcome of the differentiation process in calculus. It represents the rate at which a function's output value changes as its input value changes.
- Instantaneous rate of change: It measures the exact, instantaneous change of one quantity (the dependent variable) relative to a change in another quantity (the independent variable).
- Formal notation: Commonly denoted as ( f'(x) ), ( \frac{dy}{dx} ), or ( \frac{df(x)}{dx} ).
- Noun:
- To find the slope of the tangent line at any point on the curve, you must calculate the first derivative of the function.
- The first derivative of the position function with respect to time gives the object's velocity.
- Setting the first derivative equal to zero helps identify critical points, which can be local maxima or minima.
"First derivative test": A method in calculus used to determine whether a critical point of a function is a local maximum, local minimum, or neither, by analyzing the sign of the first derivative before and after that point.
- We applied the first derivative test to classify the critical points of the polynomial.
"First derivative of a vector function": The derivative of a vector-valued function, which results in another vector representing the instantaneous rate of change.
- The first derivative of the path vector gives the velocity vector of the particle.
Derivative (n): A more general term. The "first derivative" is a specific type of derivative. Higher-order derivatives (e.g., second derivative, third derivative) are found by repeatedly differentiating the function.
- The second derivative is found by taking the derivative of the first derivative.
Differentiation (n): The mathematical process or operation used to find a derivative.
- The first derivative is obtained through the process of differentiation.
Differential calculus (n): The branch of calculus concerned with derivatives and their applications.
- Rate of change function
- Differential coefficient (a less common, more formal term)
- Gradient (commonly used in multivariable contexts for the vector of first partial derivatives)
Take the first derivative: To perform the differentiation operation.
- The first step is to take the first derivative of the given equation.
With respect to (w.r.t.): A phrase specifying the independent variable for differentiation, crucial for defining the first derivative.
- Find the first derivative of the function *f with respect to x.*
Slope: The geometric interpretation of the first derivative at a point is the slope of the tangent line to the function's graph at that point.
- The first derivative at x=2 represents the slope of the curve at that specific point.
Instantaneous velocity: A key physical application where the first derivative of position with respect to time equals instantaneous velocity.
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the result of mathematical differentiation; the instantaneous change of one quantity relative to another; df(x)/dx
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