Basic Derivative
f(x^n)= nX^(n-1)
The derivative of a function is a measure of the rate at which the function changes with respect to its input parameter. It represents the slope of the tangent line to the curve at a given point.
To find the derivative of a function, we use the concept of limits. First, we take a small interval around the point of interest and calculate the slope of the line connecting two points within that interval. As we make this interval smaller and smaller, the slope of the line approaches the exact slope of the tangent line.
Notation for the derivative:
If y = f(x) is a function, then its derivative is denoted by f'(x) or dy/dx.
The derivative of a constant is always zero.
The derivative of a power function is given by the power multiplied by the coefficient, e.g. if y = ax^n, then y’ = anx^(n-1).
The derivative of a sum or difference of functions is the sum or difference of the derivatives of each function separately.
The chain rule is used to find the derivative of composite functions, e.g. if y = f(g(x)), then y’ = f'(g(x))g'(x).
The product rule is used to find the derivative of a product of functions, e.g. if y = f(x)g(x), then y’ = f'(x)g(x) + f(x)g'(x).
The quotient rule is used to find the derivative of a quotient of functions, e.g. if y = f(x)/g(x), then y’ = [f'(x)g(x) – f(x)g'(x)]/[g(x)]^2.
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