Slope of tangent line at a point, value of derivative at a point
Instantenous Rate of Change
When we have a function f(x), the derivative at a point, say x=a, represents the instantaneous rate of change of the function at that point. In other words, it gives us the slope of the tangent line of the function at that point. We can find the value of the derivative at x=a by taking the limit of the difference quotient of the function as h approaches 0:
f'(a) = lim[h→0] [f(a+h) – f(a)]/h
Once we find the derivative at x=a, we can use this value to find the slope of the tangent line at that point. The slope of the tangent line is simply the value of the derivative at x=a. Thus,
slope of tangent line at x=a = f'(a)
So, if we have a function f(x) = 2x^2 + 3x – 1, and we want to find the slope of the tangent line at x=2, we first find the derivative at x=2:
f'(2) = lim[h→0] [f(2+h) – f(2)]/h
f'(2) = lim[h→0] [2(2+h)^2 + 3(2+h) – 1 – (2(2)^2 + 3(2) – 1)]/h
f'(2) = lim[h→0] [2h + 10]/h = 2
Therefore, the slope of the tangent line at x=2 is 2.
More Answers:
Understanding and Addressing Decreasing Trends in Numerical Data: Strategies for Analysis and ImprovementThe Definition of Derivative in Mathematics – Understanding Instantaneous Rates of Change and Tangent Lines
Understanding the First Principle of Differentiation: Finding the Derivative of a Function