Basic Derivative
f(x^n)= nX^(n-1)
A derivative is a mathematical concept that represents the instantaneous rate of change of one variable with respect to another. In calculus, the derivative measures the slope of the tangent line to a curve at a given point.
To find the derivative of a function, we use the rules of differentiation. The basic rules for finding derivatives are as follows:
1. Power Rule: if the function is in the form f(x) = xn, then its derivative is f'(x) = nx^(n-1).
Example: if f(x) = x^3, then f'(x) = 3x^2.
2. Constant Multiple Rule: if the function is in the form f(x) = cf(x), where c is a constant, then its derivative is f'(x) = c(f(x))’.
Example: if f(x) = 2x^2, then f'(x) = 2(2x) = 4x.
3. Sum and Difference Rule: if the function is in the form f(x) = g(x) + h(x), then its derivative is f'(x) = g'(x) + h'(x).
Example: if f(x) = x^2 + 3x, then f'(x) = 2x + 3.
4. Product Rule: if the function is in the form f(x) = g(x)h(x), then its derivative is f'(x) = g'(x)h(x) + g(x)h'(x).
Example: if f(x) = x^2 * sin(x), then f'(x) = 2x*sin(x) + x^2*cos(x).
5. Quotient Rule: if the function is in the form f(x) = g(x)/h(x), then its derivative is f'(x) = [h(x)g'(x) – g(x)h'(x)]/h(x)^2.
Example: if f(x) = x/(x^2 + 1), then f'(x) = (1*(x^2 + 1) – x*(2x))/(x^2 + 1)^2.
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