Product rule for differentiation

Statement for two functions

Verbal statement

If two (possibly equal) functions are differentiable at a given real number, then their pointwise product is also differentiable at that number and the derivative of the product is the sum of two terms: the derivative of the first function times the second function and the first function times the derivative of the second function.

Statement with symbols

Suppose $f$ and $g$ are functions, both of which are differentiable at a real number $x=x_0$. Then, the product function $f\cdot g$, defined as $x\mapsto f(x)g(x)$ is also differentiable at $x$, and the derivative at $x_0$ is given as follows:

$\frac{d}{dx}[f(x)g(x)]{|}_{x=x_0}=f^{\prime }(x_0)g(x_0)+f(x_0)g^{\prime }(x_0)$

or equivalently:

$\frac{d}{dx}[f(x)g(x)]{|}_{x=x_0}=\frac{d(f(x))}{dx}{|}_{x=x_0}\cdot g(x_0)+f(x_0)\cdot \frac{d(g(x))}{dx}{|}_{x=x_0}$

If we consider the general expressions rather than evaluation at a particular point $x_0$, we can rewrite the above as:

$\frac{d}{dx}[f(x)g(x)]=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$

or equivalently:

$(f\cdot g{)}^{\prime }=(f^{\prime }\cdot g)+(f\cdot g^{\prime })$