constant multiple rule
The constant multiple rule in mathematics refers to the multiplication of a constant by a function
The constant multiple rule in mathematics refers to the multiplication of a constant by a function. It is one of the basic rules of calculus that helps simplify calculations and understand the behavior of functions.
According to the constant multiple rule, if you have a constant ‘c’ multiplied by a function f(x), you can differentiate or integrate the function by simply applying the constant ‘c’ to the derivative or integral of f(x).
In differentiation, if you have a function f(x) and a constant c, then the derivative of the constant times the function is equal to the constant multiplied by the derivative of the function. Mathematically, it can be represented as:
d/dx (c * f(x)) = c * (d/dx) f(x)
For example, if you have the function f(x) = 3x^2, and you want to find its derivative with respect to x, you can use the constant multiple rule. Since the constant ‘c’ is 3, you can simply multiply it with the derivative of x^2 which is 2x. Therefore, the derivative of 3x^2 is 6x.
In integration, the constant multiple rule is applied similarly. If you have a function f(x) and a constant c, then the integral of the constant times the function is equal to the constant multiplied by the integral of the function. Mathematically, it can be represented as:
∫ (c * f(x)) dx = c * ∫ f(x) dx
For example, if you have the function f(x) = 2x^3, and you want to find its integral with respect to x, you can use the constant multiple rule. Since the constant ‘c’ is 2, you can simply multiply it with the integral of x^3 which is (x^4)/4. Therefore, the integral of 2x^3 is (2x^4)/4 = (x^4)/2.
The constant multiple rule is a simple but powerful tool in calculus that allows us to manipulate functions when they are multiplied by constants. It helps in simplifying calculations and understanding the properties of functions.
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