{"id":230274,"date":"2025-09-10T12:08:08","date_gmt":"2025-09-10T16:08:08","guid":{"rendered":"https:\/\/ibkrcampus.com\/campus\/?p=230274"},"modified":"2025-09-10T12:08:08","modified_gmt":"2025-09-10T16:08:08","slug":"exploring-the-chain-rule-with-step-by-step-examples","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/exploring-the-chain-rule-with-step-by-step-examples\/","title":{"rendered":"Exploring the Chain Rule with Step-by-Step Examples"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><em>The article &#8220;Exploring the Chain Rule with Step-by-Step Examples&#8217; was originally posted on <a href=\"https:\/\/blog.quantinsti.com\/understanding-chain-rule\/\">QuantInsti<\/a>.<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In this blog on \u201cUnderstanding the chain rule,\u201d we will learn the math behind the application of chain rule with the help of an example.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"h-table-of-contents\"><strong>Table of Contents<\/strong><\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>What is a derivative?<\/li>\n\n\n\n<li>What is the Chain Rule?<\/li>\n\n\n\n<li>Understanding the Chain Rule<\/li>\n\n\n\n<li>Example of Chain Rule<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">For those of you who are interested in&nbsp;<a href=\"https:\/\/blog.quantinsti.com\/introduction-deep-learning-neural-network\" target=\"_blank\" rel=\"noreferrer noopener\">Neural Networks and Deep Learning<\/a>, the process of&nbsp;<a href=\"https:\/\/blog.quantinsti.com\/backpropagation\" target=\"_blank\" rel=\"noreferrer noopener\">backpropagation<\/a>&nbsp;is a very important concept which is extensively used while creating these advanced models. While performing backpropagation, we use the concept of chain rule to backpropagate the error values in prediction to adjust the weights.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To be able to understand this unit, you should know what a derivative is.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>What is a derivative?<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Don\u2019t sweat it, in case you don\u2019t know or don\u2019t remember the same, you can learn about it on the&nbsp;<a href=\"https:\/\/quantra.quantinsti.com\/glossary\" target=\"_blank\" rel=\"noreferrer noopener\">glossary section of Quantra website<\/a>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>What is the Chain Rule?<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The chain rule is basically a formula for computing the derivative of a composition of two or more functions.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Understanding the Chain Rule<\/strong><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Let us say that f and g are functions, then the chain rule expresses the derivative of their composition as<em>&nbsp;f \u2218 g<\/em>&nbsp;(the function which maps&nbsp;<em>x<\/em>&nbsp;to&nbsp;<em>f(g(x))<\/em>&nbsp;). The derivative of this composition is calculated as mentioned below.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"171\" height=\"24\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/09\/chain-rule-quantinsti-1.png\" alt=\"Chain Rule\" class=\"wp-image-230277 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 171px; aspect-ratio: 171\/24;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Here&nbsp;<em>f<\/em>&nbsp;is the function of&nbsp;<em>g<\/em>&nbsp;and&nbsp;<em>g<\/em>&nbsp;is a function of variable&nbsp;<em>x<\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Another way of writing the above rule:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"179\" height=\"24\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/09\/chain-rule-quantinsti-2.png\" alt=\"Chain Rule\" class=\"wp-image-230280 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 179px; aspect-ratio: 179\/24;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Where the function&nbsp;<em>F<\/em>&nbsp;represents the composite function&nbsp;<em>f(g(x))<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let us say that we have three variables&nbsp;<em>x, y<\/em>&nbsp;and&nbsp;<em>z<\/em>&nbsp;such that, the variable&nbsp;<em>z<\/em>&nbsp;depends on the variable&nbsp;<em>y<\/em>, which in turn depends on the variable&nbsp;<em>x<\/em>. So&nbsp;<em>y<\/em>&nbsp;and&nbsp;<em>z<\/em>&nbsp;are dependent variables, and&nbsp;<em>z<\/em>, via the intermediate variable of&nbsp;<em>y<\/em>, depends on&nbsp;<em>x<\/em>. Then the chain rule for differentiating the variable&nbsp;<em>z<\/em>&nbsp;may be written in the following manner.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"123\" height=\"47\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/09\/chain-rule-quantinsti-3.png\" alt=\"Chain Rule\" class=\"wp-image-230281 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 123px; aspect-ratio: 123\/47;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">This is the final formula that we use in backpropagation.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Here&nbsp;<em>z<\/em>&nbsp;is the function of&nbsp;<em>y<\/em>,<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em><strong>z = f(y)<\/strong><\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and&nbsp;<em>y<\/em>&nbsp;is a function of&nbsp;<em>x<\/em>,<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em><strong>y= g(x)<\/strong><\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Using the previous formula, we can rewrite the differential equation as follows:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"356\" height=\"47\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/09\/chain-rule-quantinsti-4.png\" alt=\"Chain Rule\" class=\"wp-image-230283 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/09\/chain-rule-quantinsti-4.png 356w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/09\/chain-rule-quantinsti-4-300x40.png 300w\" data-sizes=\"(max-width: 356px) 100vw, 356px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 356px; aspect-ratio: 356\/47;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Let us understand this better with the help of an example.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example of Chain Rule<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let us understand the chain rule with the help of a well-known example from Wikipedia. Assume that you are falling from the sky, the atmospheric pressure keeps changing during the fall. Check out the graph below to understand this change.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"710\" height=\"470\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/09\/chart-chain-rule.jpg\" alt=\"Chain Rule\" class=\"wp-image-230284 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/09\/chart-chain-rule.jpg 710w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/09\/chart-chain-rule-700x463.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/09\/chart-chain-rule-300x199.jpg 300w\" data-sizes=\"(max-width: 710px) 100vw, 710px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 710px; aspect-ratio: 710\/470;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">At the time of your fall, 4000 meters above sea level, the initial velocity was zero, and the gravity is 9.8 meters per second squared. Now compare this situation to the previous chain rule equation. Let us say that the variable&nbsp;<em>x<\/em>&nbsp;in the equation is variable&nbsp;<em>t<\/em>, or time.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Then the variable&nbsp;<em>y<\/em>&nbsp;or&nbsp;<em>g(t)<\/em>, which is the distance travelled by you since the beginning of the fall is given by<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em><strong>g(t) = 0.5*9.8t<sup>2<\/sup><\/strong><\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So, the height from the mean sea level can be given by the variable&nbsp;<em>h<\/em>, which is<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em><strong>h = 4000 &#8211; g(t)<\/strong><\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let us say that we also know, based on a model, the atmospheric pressure at a height&nbsp;<em>h<\/em>&nbsp;as:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em><strong>f(h) = 101325 e<sup>\u22120.0001h<\/sup><\/strong><\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">These two equations can be differentiated by their respective variable to get the following information:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em><strong>g\u2032(t) = \u22129.8t<\/strong><\/em>,<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where,&nbsp;<em><strong>g\u2032(t)&nbsp;<\/strong><\/em>is the velocity of you at time&nbsp;<em>t<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em><strong>f\u2032(h) = \u221210.1325<sup>e\u22120.0001h<\/sup><\/strong><\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">where,&nbsp;<em><strong>f\u2032(h)<\/strong><\/em>&nbsp;is the rate of change in atmospheric pressure with respect to height&nbsp;<em>h<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now let us understand how we can combine these two equations to derive the<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">the rate of change in the atmospheric pressure with respect to time at&nbsp;<em>t<\/em>&nbsp;seconds after the skydiver&#8217;s jump, using the chain rule:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"218\" height=\"24\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/09\/chain-rule-quantinsti-5.png\" alt=\"Chain Rule\" class=\"wp-image-230287 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 218px; aspect-ratio: 218\/24;\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"408\" height=\"29\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/09\/chain-rule-quantinsti-7.png\" alt=\"Chain Rule\" class=\"wp-image-230300 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/09\/chain-rule-quantinsti-7.png 408w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/09\/chain-rule-quantinsti-7-300x21.png 300w\" data-sizes=\"(max-width: 408px) 100vw, 408px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 408px; aspect-ratio: 408\/29;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">This equation gives us the rate of change of atmospheric pressure with respect to time since fall. In neural networks, we will need to calculate the change in weights at each neuron with respect to the errors in prediction. As you might have imagined by now, the chain rule helps adjusts these weights accordingly.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Conclusion<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If we want to apply the chain rule to backpropagate the error in neural networks, then we will be using an equation such as this.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"455\" height=\"70\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/09\/chain-rule-quantinsti-6.png\" alt=\"Chain Rule\" class=\"wp-image-230290 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/09\/chain-rule-quantinsti-6.png 455w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2025\/09\/chain-rule-quantinsti-6-300x46.png 300w\" data-sizes=\"(max-width: 455px) 100vw, 455px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 455px; aspect-ratio: 455\/70;\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">In the Quantra\u2019s course on&nbsp;<a href=\"https:\/\/quantra.quantinsti.com\/course\/neural-networks-deep-learning-trading-ernest-chan\" target=\"_blank\" rel=\"noreferrer noopener\">Deep Learning in Trading with Dr. E. P. Chan<\/a>, we will help you not only understand advanced concepts such as deep learning, but also apply them in the context of trading.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Disclaimer:&nbsp;All investments and trading in the stock market involve risk. Any decisions to place trades in the financial markets, including trading in stock or options or other financial instruments is a personal decision that should only be made after thorough research, including a personal risk and financial assessment and the engagement of professional assistance to the extent you believe necessary. The trading strategies or related information mentioned in this article is for informational purposes only.<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Suggested Reads:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/blog.quantinsti.com\/forward-propagation-neural-networks\" target=\"_blank\" rel=\"noreferrer noopener\">Forward Propagation In Neural Networks<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/blog.quantinsti.com\/backpropagation\" target=\"_blank\" rel=\"noreferrer noopener\">Understanding Backpropagation<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>The chain rule is basically a formula for computing the derivative of a composition of two or more functions.<\/p>\n","protected":false},"author":134,"featured_media":182721,"comment_status":"open","ping_status":"closed","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[339,338,341],"tags":[20548,2105,1006,15593],"contributors-categories":[13654],"class_list":["post-230274","post","type-post","status-publish","format-standard","has-post-thumbnail","category-data-science","category-ibkr-quant-news","category-quant-development","tag-chain-rule","tag-deep-learning","tag-fintech","tag-neural-networks","contributors-categories-quantinsti"],"pp_statuses_selecting_workflow":false,"pp_workflow_action":"current","pp_status_selection":"publish","acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.9 (Yoast SEO v28.4) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Exploring the Chain Rule with Step-by-Step Examples<\/title>\n<meta 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