The Chain Rule: Learn It 4

The Chain Rule Using Leibniz’s Notation

As with other derivatives that we have seen, we can express the chain rule using Leibniz’s notation. This notation for the chain rule is used heavily in physics applications.

For [latex]h(x)=f(g(x))[/latex], let [latex]u=g(x)[/latex] and [latex]y=h(x)=g(u)[/latex]. Thus,

[latex]h^{\prime}(x)=\frac{dy}{dx}, \, f^{\prime}(g(x))=f^{\prime}(u)=\frac{dy}{du}[/latex], and [latex]g^{\prime}(x)=\frac{du}{dx}[/latex]

Consequently,

[latex]\frac{dy}{dx}=h^{\prime}(x)=f^{\prime}(g(x))g^{\prime}(x)=\frac{dy}{du} \cdot \frac{du}{dx}[/latex]

chain rule using Leibniz’s notation

If [latex]y[/latex] is a function of [latex]u[/latex], and [latex]u[/latex] is a function of [latex]x[/latex], then

[latex]\dfrac{dy}{dx}=\dfrac{dy}{du} \cdot \dfrac{du}{dx}[/latex]

Find the derivative of [latex]y=\left(\dfrac{x}{3x+2}\right)^5[/latex]

Find the derivative of [latex]y= \tan (4x^2-3x+1)[/latex]