derivitive of a function f at a number c
To find the derivative of a function f at a number c, we can use the concept of the derivative being the slope of the tangent line to the graph of the function at that point
To find the derivative of a function f at a number c, we can use the concept of the derivative being the slope of the tangent line to the graph of the function at that point.
The derivative of a function f(x) at a given number c, denoted by f'(c), can be calculated using the following formula:
f'(c) = lim (h→0) [f(c + h) – f(c)] / h
Where h represents a small change in x as it approaches zero.
To find the derivative at a specific number c, we follow these steps:
1. Substitute c + h into the function for x: f(c + h)
2. Subtract the function value at c: f(c)
3. Divide the result by h
4. Take the limit as h approaches 0
Let’s understand this with an example. Consider the function f(x) = x^2 and suppose we want to find the derivative at the number c = 3.
Step 1: Substitute c + h into the function for x: f(c + h) = (c + h)^2
f(c + h) = (3 + h)^2 = 9 + 6h + h^2
Step 2: Subtract the function value at c: f(c) = 3^2 = 9
Step 3: Divide the result by h:
[f(c + h) – f(c)] / h = [9 + 6h + h^2 – 9] / h
= (6h + h^2) / h
= 6 + h
Step 4: Take the limit as h approaches 0:
lim (h→0) (6 + h) = 6
So, the derivative of f(x) = x^2 at the number c = 3 is 6.
In general, the process to find the derivative at a specific number c follows the same steps, involving substitution, algebraic manipulation, and taking the limit.
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