What is 'Chain Rule' and why is it useful?

The chain rule is most commonly seen in Leibniz's notation, which is as follows:<o:p></o:p>

 

dz/dx = dz/dy * dy/dx<o:p></o:p>

 

You can remember it intuitively by thinking of the two 'dy' terms cancelling to leave dz/dx.<o:p></o:p>

 

So why use the chain rule?<o:p></o:p>

You are used to differentiating equations in the form y = f(x), but say both sides of the equation where functions eg g(y) = f(x) and you had to differentiate the equation with respect to x. <o:p></o:p>

g is a function of y, not x, so you can't simply calculate dg(y)/dx like you can df(x)/dx. Using the chain rule we can express dg(y)/dx as dg(y)/dy * dy/dx. These two terms can be calculated (assuming y is a function of x). This is really what the chain rule is saying: that the derivative of a function composition can be expressed as a product of the respective derivatives.<o:p></o:p>

 

Another example of when the chain rule might come in useful is in mechanics: Acceleration is defined as the derivative of velocity: dv/dt. Sometimes though it might be useful to integrate acceleration of a distance, x, rather than over time. To eliminate time from this expression we can use the chain rule by saying dv/dt = dv/dx * dx/dt. Then noting that dx/dt is in fact velocity (v = dx/dt) we can write that dv/dt = v * dv/dx thus making acceleration a function only of velocity and position.<o:p></o:p>