The derivative of a power function is a function in which the power on [latex]x[/latex] becomes the coefficient of the term and the power on [latex]x[/latex] in the derivative decreases by 1.
The derivative of a constant [latex]c[/latex] multiplied by a function [latex]f[/latex] is the same as the constant multiplied by the derivative.
The derivative of the sum of a function [latex]f[/latex] and a function [latex]g[/latex] is the same as the sum of the derivative of [latex]f[/latex] and the derivative of [latex]g[/latex].
The derivative of the difference of a function [latex]f[/latex] and a function [latex]g[/latex] is the same as the difference of the derivative of [latex]f[/latex] and the derivative of [latex]g[/latex].
The derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function.
The derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function.
We used the limit definition of the derivative to develop formulas that allow us to find derivatives without resorting to the definition of the derivative. These formulas can be used singly or in combination with each other.
Glossary
constant multiple rule
the derivative of a constant [latex]c[/latex] multiplied by a function [latex]f[/latex] is the same as the constant multiplied by the derivative: [latex]\frac{d}{dx}(cf(x))=cf^{\prime}(x)[/latex]
constant rule
the derivative of a constant function is zero: [latex]\frac{d}{dx}(c)=0[/latex], where [latex]c[/latex] is a constant
difference rule
the derivative of the difference of a function [latex]f[/latex] and a function [latex]g[/latex] is the same as the difference of the derivative of [latex]f[/latex] and the derivative of [latex]g[/latex]: [latex]\frac{d}{dx}(f(x)-g(x))=f^{\prime}(x)-g^{\prime}(x)[/latex]
power rule
the derivative of a power function is a function in which the power on [latex]x[/latex] becomes the coefficient of the term and the power on [latex]x[/latex] in the derivative decreases by 1: If [latex]n[/latex] is an integer, then [latex]\frac{d}{dx}(x^n)=nx^{n-1}[/latex]
product rule
the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function: [latex]\frac{d}{dx}(f(x)g(x))=f^{\prime}(x)g(x)+g^{\prime}(x)f(x)[/latex]
quotient rule
the derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function: [latex]\frac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)=\dfrac{f^{\prime}(x)g(x)-g^{\prime}(x)f(x)}{(g(x))^2}[/latex]
sum rule
the derivative of the sum of a function [latex]f[/latex] and a function [latex]g[/latex] is the same as the sum of the derivative of [latex]f[/latex] and the derivative of [latex]g[/latex]: [latex]\frac{d}{dx}(f(x)+g(x))=f^{\prime}(x)+g^{\prime}(x)[/latex]
