Question

State the chain rule for differentiating a composition $g\left(h\right(x\left)\right)$of two functions expressed

(a) in “prime” notation and

(b) in Leibniz notation.

Short answer

a) In prime notation the derivative is $\left(g\circ h\right)\text{'}\left(x\right)=g\text{'}\left(h\right(x\left)\right)\times h\text{'}\left(x\right)$

b) In Leibniz notation the derivative is:$\frac{dg}{dx}=\frac{dg}{dh}\times \frac{dh}{dx}$

Step-by-step solution

Step 1. Given information:

The composite function is:

$g\left(h\right(x\left)\right)$

Part (a). Step 1. The derivative in prime notation:

In prime notation the derivative of the function is written as:

$(g\circ h)\text{'}\left(x\right)=g\text{'}\left(h\right(x\left)\right)\times h\text{'}\left(x\right)$

Part (b). Step 1. The derivative in Leibniz notation:

In Leibniz notation, the derivative is:

$\frac{dg}{dx}=\frac{dg}{dh}\times \frac{dh}{dx}$