find the slope at given value AKA
instantaneous
To find the slope at a given value, we need to take the derivative of the function at that point.
The derivative of a function gives us the slope of the tangent line to the function at any given point.
Let’s say we have a function f(x) and we want to find the slope at a point x=a.
1. First, we need to take the derivative of the function f(x) with respect to x.
f'(x) = df/dx
2. Then, we substitute x=a in the derivative we just found:
f'(a) = df/dx | x=a
3. Finally, we evaluate the derivative at x=a to find the slope at that point.
f'(a) gives us the slope of the tangent line to the function f(x) at x=a.
For example, if we have a function f(x) = 2x^2 + 3x + 1 and we want to find the slope at x=2, we would do the following:
1. Take the derivative of f(x) with respect to x:
f'(x) = 4x + 3
2. Substitute x=2 in our derivative:
f'(2) = 4(2) + 3 = 11
3. Evaluate the derivative at x=2 to find the slope at that point:
The slope of the tangent line to f(x) at x=2 is 11.
Therefore, the slope at a given value can be found by taking the derivative of the function and evaluating it at the given point.
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