Question

Write down a rule for differentiating a composition $f\left(u\right(v\left(w\right(x\left)\right)\left)\right)$of four functions

(a) in “prime” notation and

(b) in Leibniz notation.

Short answer

(a) in “prime” notation the derivative is:$(f\circ u\circ v\circ w)\text{'}\left(x\right)=f\text{'}(u\circ v\circ w)\left(x\right)\times u\text{'}(v\circ w)\left(x\right)\times v\text{'}\left(w\right(x\left)\right)\times w\text{'}\left(x\right)$

(b) in Leibniz notation the derivative is:$\frac{df}{dx}=\frac{df}{du}\times \frac{du}{dv}\times \frac{dv}{dw}\times \frac{dw}{dx}$

Step-by-step solution

Step 1. Given information:

The composite function is;

$f\left(u\right(v\left(w\right(x\left)\right)\left)\right)$

Part (a). Step 1. Derivative in prime notation:

The derivative of the given function in prime notation is;$(f\circ u\circ v\circ w)\text{'}\left(x\right)=f\text{'}(u\circ v\circ w)\left(x\right)\times u\text{'}(v\circ w)\left(x\right)\times v\text{'}\left(w\right(x\left)\right)\times w\text{'}\left(x\right)$

Part (b). Step 1. Derivative in Leibniz notation:

In Leibniz notation the derivative of the function is:

$\frac{df}{dx}=\frac{df}{du}\times \frac{du}{dv}\times \frac{dv}{dw}\times \frac{dw}{dx}$