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This calculus math tutorial from NutshellMath offers homework help on understanding the concept of derivatives. As presented in the tutorial, the derivative of a function at a point a, denoted as f’(a), is defined as the limit as h approaches zero of (f(a+h)-f(a))/(h-a). This definition is very similar to the definition of the slope of a line between two points. As the two points in the slope formula near each other, which occurs as h approaches zero, the slope will approach the derivative of the function at that point. From this relationship, it is possible to see how the derivative is considered the slope of a line tangent to a function at the chosen point. A line drawn at the point f(a) with the slope f’(a) will be tangent to the function f(x). Using the derivative of a function as the slope of a tangent line is one of the major applications of the derivative.
A second major application of the derivative is as a measure of instantaneous rate of change. The slope of a linear function is the constant rate of change of y with respect to x. For non-linear functions, the slope of the line will change with respect to x. The derivative of a function with respect to x at any point is the instantaneous rate of change of the function at that point. The most common example of this is using a position function, with an independent variable t for time, and a position variable f(t) as the dependent variable. The derivative of f(t) with respect to t will yield the instantaneous rate of change in position over time, which is the velocity. If the relationship between f(t) and t is linear, then the slope will be constant, and the velocity will be constant. If the relationship is not linear, then the derivative will vary with respect to t, and the velocity will change over time. Understanding that the derivative of a position function is a velocity function is very important in physics and other areas of science.
This introduction to derivatives should reinforce the concept of a derivative as both the slope of a line tangent to a curve, and the instantaneous rate of change of a function.
This calculus math tutorial from NutshellMath offers homework help on understanding the concept of derivatives. As presented in the tutorial, the derivative of a function at a point a, denoted as f’(a), is defined as the limit as h approaches zero of (f(a+h)-f(a))/(h-a). This definition is very similar to the definition of the slope of a line between two points. As the two points in the slope formula near each other, which occurs as h approaches zero, the slope will approach the derivative of the function at that point. From this relationship, it is possible to see how the derivative is considered the slope of a line tangent to a function at the chosen point. A line drawn at the point f(a) with the slope f’(a) will be tangent to the function f(x). Using the derivative of a function as the slope of a tangent line is one of the major applications of the derivative.
A second major application of the derivative is as a measure of instantaneous rate of change. The slope of a linear function is the constant rate of change of y with respect to x. For non-linear functions, the slope of the line will change with respect to x. The derivative of a function with respect to x at any point is the instantaneous rate of change of the function at that point. The most common example of this is using a position function, with an independent variable t for time, and a position variable f(t) as the dependent variable. The derivative of f(t) with respect to t will yield the instantaneous rate of change in position over time, which is the velocity. If the relationship between f(t) and t is linear, then the slope will be constant, and the velocity will be constant. If the relationship is not linear, then the derivative will vary with respect to t, and the velocity will change over time. Understanding that the derivative of a position function is a velocity function is very important in physics and other areas of science.
This introduction to derivatives should reinforce the concept of a derivative as both the slope of a line tangent to a curve, and the instantaneous rate of change of a function.