Product and Quotient Rules

Calculus Notes

Section 2.3: Product and Quotient Rules and Higher-Order Derivatives

The product rule states that $$\boxed{\frac{\diff}{\diff x}\left[f(x)g(x)\right]=f'(x)g(x) + g'(x)f(x)}$$

"The derivative of a product is the derivative of the first times the second plus the derivative of the second times the first."

This equation is sometimes abbreviated as: $$(fg)' = f'g +g'f $$ The quotient rule states that $$\boxed{\frac{\diff}{\diff x}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - g'(x)f(x)}{\left[g(x)\right]^2}}$$

“The derivative of a quotient is the derivative of the numerator time the numerator minus the derivative of the denominator times the numerator all over the denominator squared.”

In other words,

$$(f/g)' = \frac{f'g-g'f}{g^2}$$

Because many functions can be written as the product or quotient of simpler functions, the product and quotient rules vastly increase the number of functions that can be differentiated.

Higher-order derivatives

To calculate the second derivative of a function, first take its derivative, and then take its derivative again. If $y = f(x)$, the second derivative can be written in the following ways: $$ y'' = f''(x) = \frac{\diff^2 y}{\diff x^2} = \frac{\diff^2 f}{\diff x^2} = \frac{\diff^2 }{\diff x^2} f(x) = D_x^2 f(x) = D_{xx} f(x)$$ The process of differentiation can be repeated an indefinite number of times. In general, the $n$th derivative is expressed as follows: $$y^{(n)} = f^{(n)}(x) = \frac{\diff^n y}{\diff x^n} = \frac{\diff^n f}{\diff x^n} = \frac{\diff^n}{\diff x^n} f(x) = D^n_x f(x)$$ One familiar second derivative is acceleration, which is the time derivative of the velocity, and the second time derivative of displacement. $$ a = \frac{\diff v}{\diff x} = \frac{\diff^2 s}{\diff t^2}$$