derivative of a function at a point
The derivative of a function at a point measures the rate of change of the function at that specific point
The derivative of a function at a point measures the rate of change of the function at that specific point. In other words, it tells us how fast the function is changing at that point.
The derivative of a function f(x) at a point x = a is denoted as f'(a) or dy/dx evaluated at x = a. There are several ways to calculate the derivative at a point, depending on the form of the function itself.
1. Power Rule: If the function is in the form f(x) = x^n, where n is a constant, the derivative can be found using the power rule. The derivative is given by f'(x) = n * x^(n-1). For example, if f(x) = x^2, then f'(x) = 2x.
2. Constant Rule: If the function is a constant, such as f(x) = c, where c is a constant, the derivative is always zero. This is because a constant value does not change, so the rate of change is zero.
3. Sum and Difference Rule: If the function is the sum or difference of two functions, f(x) = g(x) ± h(x), then the derivative is the sum or difference of the derivatives of the individual functions. In other words, f'(x) = g'(x) ± h'(x).
4. Product Rule: If the function is the product of two functions, f(x) = g(x) * h(x), then the derivative can be found using the product rule. The derivative is given by f'(x) = g'(x) * h(x) + g(x) * h'(x).
5. Quotient Rule: If the function is the quotient of two functions, f(x) = g(x) / h(x), then the derivative can be found using the quotient rule. The derivative is given by f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / h(x)^2.
6. Chain Rule: If the function is composed of multiple functions, f(x) = g(h(x)), then the derivative can be found using the chain rule. The derivative is given by f'(x) = g'(h(x)) * h'(x).
To find the derivative at a specific point, simply substitute the value of x into the derivative expression you determined using one of the above rules. This will give you the rate of change (slope) of the function at that point.
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