{"id":376,"date":"2021-02-04T01:19:06","date_gmt":"2021-02-04T01:19:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=376"},"modified":"2022-03-16T05:35:02","modified_gmt":"2022-03-16T05:35:02","slug":"the-chain-rule-using-leibnizs-notation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/the-chain-rule-using-leibnizs-notation\/","title":{"raw":"The Chain Rule Using Leibniz\u2019s Notation","rendered":"The Chain Rule Using Leibniz\u2019s Notation"},"content":{"raw":"

As with other derivatives that we have seen, we can express the chain rule using Leibniz\u2019s notation. This notation for the chain rule is used heavily in physics applications.<\/p>\r\n

For [latex]h(x)=f(g(x))[\/latex], let [latex]u=g(x)[\/latex] and [latex]y=h(x)=g(u)[\/latex]. Thus,<\/p>\r\n\r\n

[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]<\/div>\r\n \r\n

Consequently,<\/p>\r\n\r\n

[latex]\\frac{dy}{dx}=h^{\\prime}(x)=f^{\\prime}(g(x))g^{\\prime}(x)=\\frac{dy}{du} \\cdot \\frac{du}{dx}[\/latex]<\/div>\r\n \r\n
<\/div>\r\n
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Chain Rule Using Leibniz\u2019s Notation<\/h3>\r\n\r\n
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If [latex]y[\/latex] is a function of [latex]u[\/latex], and [latex]u[\/latex] is a function of [latex]x[\/latex], then<\/p>\r\n\r\n

[latex]\\dfrac{dy}{dx}=\\dfrac{dy}{du} \\cdot \\dfrac{du}{dx}[\/latex]<\/div>\r\n \r\n\r\n<\/div>\r\n
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Example: Taking a Derivative Using Leibniz\u2019s Notation, 1<\/h3>\r\n

Find the derivative of [latex]y=\\left(\\dfrac{x}{3x+2}\\right)^5[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169739264248\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169739264248\"]\r\n

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,<\/p>\r\n\r\n

[latex]\\frac{du}{dx}=\\frac{2}{(3x+2)^2}[\/latex]<\/div>\r\n \r\n

and<\/p>\r\n\r\n

[latex]\\frac{dy}{du}=5u^4[\/latex]<\/div>\r\n \r\n

Finally, we put it all together.<\/p>\r\n\r\n

[latex]\\begin{array}{lllll}\\frac{dy}{dx} & =\\frac{dy}{du} \\cdot \\frac{du}{dx} & & & \\text{Apply the chain rule.} \\\\ & =5u^4 \\cdot \\frac{2}{(3x+2)^2} & & & \\text{Substitute} \\, \\frac{dy}{du}=5u^4 \\, \\text{and} \\, \\frac{du}{dx}=\\frac{2}{(3x+2)^2}. \\\\ & =5(\\frac{x}{3x+2})^4 \\cdot \\frac{2}{(3x+2)^2} & & & \\text{Substitute} \\, u=\\frac{x}{3x+2}. \\\\ & =\\frac{10x^4}{(3x+2)^6} & & & \\text{Simplify.} \\end{array}[\/latex]<\/div>\r\n \r\n

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.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

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Example: Taking a Derivative Using Leibniz\u2019s Notation, 2<\/h3>\r\n

Find the derivative of [latex]y= \\tan (4x^2-3x+1)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1169736661260\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736661260\"]\r\n

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]:<\/p>\r\n\r\n

[latex]\\frac{du}{dx}=8x-3[\/latex] and [latex]\\frac{dy}{du}=\\sec^2 u[\/latex].<\/div>\r\n \r\n

Finally, we put it all together.<\/p>\r\n\r\n

[latex]\\begin{array}{lllll}\\frac{dy}{dx} & =\\frac{dy}{du} \\cdot \\frac{du}{dx} & & & \\text{Apply the chain rule.} \\\\ & = \\sec^2 u \\cdot (8x-3) & & & \\text{Use} \\, \\frac{du}{dx}=8x-3 \\, \\text{and} \\, \\frac{dy}{du}= \\sec^2 u. \\\\ & = \\sec^2 (4x^2-3x+1) \\cdot (8x-3) & & & \\text{Substitute} \\, u=4x^2-3x+1. \\end{array}[\/latex]<\/div>\r\n \r\n
[\/hidden-answer]<\/div>\r\n<\/div>\r\n
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Try It<\/h3>\r\n

Use Leibniz\u2019s 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].<\/p>\r\n[reveal-answer q=\"844612\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"844612\"]\r\n

Let [latex]u=x^3[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"fs-id1169736594031\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1169736594031\"]\r\n

[latex]\\frac{dy}{dx}=-3x^2 \\sin (x^3)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to the above Try It.\r\n\r\n