Differentiation of a function of a function

It sometimes happens that $ y$, instead of being defined directly as a function of $ x$, is given as a function of another variable $ v$, which is defined as a function of $ x$. In that case $ y$ is a function of $ x$ through $ v$ and is called a function of a function or a composite function. The process of substituting one function into another is sometimes called composition.

For example, if $ y = \frac{2v}{1 - v^2}$, and $ v = 1-x^2$, then $ y$ is a function of a function. By eliminating $ v$ we may express $ y$ directly as a function of $ x$, but in general this is not the best plan when we wish to find $ \frac{dy}{dx}$.

If $ y = f(v)$ and $ v = g (x)$, then $ y$ is a function of $ x$ through $ v$. Hence, when we let $ x$ take on an increment $ \Delta x$, $ v$ will take on an increment $ \Delta v$ and $ y$ will also take on a corresponding increment $ \Delta y$. Keeping this in mind, let us apply the General Rule simultaneously to the two functions $ y = f(v)$ and $ v = g (x)$.

The above formula is sometimes referred to as the chain rule for differentiation. If $ y = f(v)$ and $ v = g (x)$, the derivative of $ y$ with respect to $ x$ equals the product of the derivative of $ y$ with respect to $ v$ and the derivative of $ v$ with respect to $ x$.

david joyner 2008-11-22