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]
Example: Taking a Derivative Using Leibniz’s Notation, 1
Find the derivative of [latex]y=\left(\dfrac{x}{3x+2}\right)^5[/latex]
Show Solution
First, let [latex]u=\frac{x}{3x+2}[/latex]. Thus, [latex]y=u^5[/latex]. Next, find [latex]\frac{du}{dx}[/latex] and [latex]\frac{dy}{du}[/latex]. Using the quotient rule,
It is important to remember that, when using the Leibniz form of the chain rule, the final answer must be expressed entirely in terms of the original variable given in the problem.
Example: Taking a Derivative Using Leibniz’s Notation, 2
Find the derivative of [latex]y= \tan (4x^2-3x+1)[/latex]
Show Solution
First, let [latex]u=4x^2-3x+1[/latex]. Then [latex]y= \tan u[/latex]. Next, find [latex]\frac{du}{dx}[/latex] and [latex]\frac{dy}{du}[/latex]:
[latex]\frac{du}{dx}=8x-3[/latex] and [latex]\frac{dy}{du}=\sec^2 u[/latex].
Use Leibniz’s notation to find the derivative of [latex]y= \cos (x^3)[/latex]. Make sure that the final answer is expressed entirely in terms of the variable [latex]x[/latex].
Hint
Let [latex]u=x^3[/latex].
Show Solution
[latex]\frac{dy}{dx}=-3x^2 \sin (x^3)[/latex]
Watch the following video to see the worked solution to the above Try It.
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