Summary <\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n\n\n\n\nAccording to Bright Storm, the definition of the derivative is:<\/p>\n\n\n\n
In calculus, the slope of a line tangent to the curve at a specific point. In other words, the instantaneous rate of change of the limits of the functions as time approaches zero among the measurements is said to be the derivative of the function.<\/p>\n\n\n\n
The derivative can be calculated for many types of functions, such as constant, linear, power, exponential, polynomial, or logarithmic functions. It calculates the derivatives of these functions with respect to their independent variables, so it is also known as a differential.<\/p>\n\n\n\n
The notation used to differentiate the function is d\/dx, where x can be changed with any independent variable. The limits in calculus are widely used to define the differentiation of functions.<\/p>\n\n\n\n
d\/dx g(x) = lim<\/strong>h\u21920<\/sub><\/strong> (g (x + h) \u2013 g(x)) \/ h<\/strong><\/p>\n\n\n\n\nd\/dx is a differential notation.<\/li>\n\n\n\n Lime is the limit notation.<\/li>\n\n\n\n h is a specific point.<\/li>\n\n\n\n g(x) is the given function.<\/li>\n<\/ul>\n\n\n\n<\/span>Rules of differentiation<\/strong><\/span><\/h3>\n\n\n\nThere are various rules of differentiation used to differentiate the function corresponding to the independent variables.<\/p>\n\n\n\n
\nSum rule:<\/strong> d\/dx [g(x) + h(x)] = d\/dx g(x) + d\/dx h(x)<\/li>\n\n\n\nDifference rule:<\/strong> d\/dx [g(x) – h(x)] = d\/dx g(x) – d\/dx h(x)<\/li>\n\n\n\nConstant rule:<\/strong> d\/dx [A] = 0, where A is any constant<\/li>\n\n\n\nConstant function rule:<\/strong> d\/dx [Af(x)] = Ad\/dx [f(x)], where A is any constant<\/li>\n\n\n\nPower rule:<\/strong> d\/dx [f(x)] n<\/sup> = n[f(x)] n-1<\/sup><\/li>\n\n\n\nProduct rule:<\/strong> d\/dx [g(x) * h(x)] = h(x) * [d\/dx g(x)] + g(x) * [d\/dx h(x)]<\/li>\n\n\n\nQuotient rule:<\/strong> d\/dx [g(x) \/ h(x)] = 1\/(h(x))2<\/sup> [h(x) * [d\/dx g(x)] – g(x) * [d\/dx h(x)]]<\/li>\n\n\n\nChain rule: <\/strong>dy\/dx= [dy\/du * du\/dx]<\/li>\n<\/ul>\n\n\n\n<\/span>How to calculate derivatives by using the chain rule?<\/strong><\/span><\/h3>\n\n\n\nThe problems of derivatives can be solved easily by using the rules of differentiation. The following are a few examples of derivatives solved by using the chain rule. Keep one thing in mind: to apply the chain rule, you also need to use the other rules of differentiation.<\/p>\n\n\n\n
Example 1<\/strong><\/p>\n\n\n\nFind the derivative of 3x3<\/sup> + 4x2<\/sup> + sin(x) + 34 with respect to x.<\/p>\n\n\n\nSolution <\/strong><\/p>\n\n\n\nStep 1:<\/strong> Use the differential notation to write the given function.<\/p>\n\n\n\nd\/dx [3x3<\/sup> + 4x2<\/sup> + sin(x) + 34]<\/p>\n\n\n\nStep 2:<\/strong> Apply the sum rule of differentiation and write the derivative notation with each function separately.<\/p>\n\n\n\nd\/dx [3x3<\/sup> + 4x2<\/sup> + sin(x) + 34] = d\/dx [3x3<\/sup>] + d\/dx [4x2<\/sup>] + d\/dx [sin(x)] + d\/dx [34]<\/p>\n\n\n\nStep 3:<\/strong> Apply the constant and the constant function rule.<\/p>\n\n\n\nd\/dx [3x3<\/sup> + 4x2<\/sup> + sin(x) + 34] = 3d\/dx [x3<\/sup>] + 4d\/dx [x2<\/sup>] + d\/dx [sin(x)] + 0<\/p>\n\n\n\n = 3d\/dx [x3<\/sup>] + 4d\/dx [x2<\/sup>] + d\/dx [sin(x)]<\/p>\n\n\n\nStep 4:<\/strong> Now use the power rule of differentiation d\/dx [f(x)] n<\/sup> = n[f(x)] n-1<\/sup> where n = 2, 3.<\/p>\n\n\n\nd\/dx [3x3<\/sup> + 4x2<\/sup> + sin(x) + 34] = 3 [3x3-1<\/sup>] + 4 [2x2-1<\/sup>] + d\/dx [sin(x)]<\/p>\n\n\n\n = (3 * 3) x3-1<\/sup> + (4 * 2) x2-1<\/sup> + d\/dx [sin(x)]<\/p>\n\n\n\n = 9x2<\/sup> + 8x1<\/sup> + d\/dx [sin(x)]<\/p>\n\n\n\n = 9x2<\/sup> + 8x + d\/dx [sin(x)]<\/p>\n\n\n\nStep 5:<\/strong> Use the chain rule to find the differential of sin(x).<\/p>\n\n\n\nd\/dx sin(x) = d\/du sin(u) * du\/dx, where u = x<\/p>\n\n\n\n
d\/dx [3x3<\/sup> + 4x2<\/sup> + sin(x) + 34] = 9x2<\/sup> + 8x + cos(x) d\/dx [x]<\/p>\n\n\n\n = 9x2<\/sup> + 8x + cos(x) [1]<\/p>\n\n\n\n = 9x2<\/sup> + 8x + cos(x)<\/p>\n\n\n\n = x (9x + 8) + cos(x)<\/p>\n\n\n\n
You can also get the step-by-step solution to differential calculus problems to avoid lengthy calculations by using a derivative calculator. Follow the steps below to use this calculator.<\/p>\n\n\n\n
Step 1:<\/strong> Input the function.<\/p>\n\n\n\nStep 2:<\/strong> Select the corresponding variable.<\/p>\n\n\n\nStep 3:<\/strong> Write the order of derivative e.g., 1 for the first derivative.<\/p>\n\n\n\nStep 4:<\/strong> Hit the calculate button.<\/p>\n\n\n\nStep 5:<\/strong> The step-by-step solution of the given function will show below the calculate button in a couple of seconds.<\/p>\n\n\n\nExample 2<\/strong><\/p>\n\n\n\nFind the derivative of 13x2<\/sup> – 14x – 4cos(x) – 54 with respect to x.<\/p>\n\n\n\nSolution <\/strong><\/p>\n\n\n\nStep 1:<\/strong> Use the differential notation to write the given function.<\/p>\n\n\n\nd\/dx [13x2<\/sup> – 14x – 4cos(x) – 54]<\/p>\n\n\n\nStep 2:<\/strong> Apply the difference rule of differentiation and write the derivative notation with each function separately.<\/p>\n\n\n\nd\/dx [13x2<\/sup> – 14x – 4cos(x) – 54] = d\/dx [13x2<\/sup>] – d\/dx [14x] – d\/dx [cos(x)] – d\/dx [54]<\/p>\n\n\n\nStep 3:<\/strong> Apply the constant and the constant function rule.<\/p>\n\n\n\nd\/dx [13x2<\/sup> – 14x – 4cos(x) – 54] = 13d\/dx [x2<\/sup>] – 14d\/dx [x] – 4d\/dx [cos(x)] – 0<\/p>\n\n\n\n = 13d\/dx [x2<\/sup>] – 14d\/dx [x] – 4d\/dx [cos(x)] <\/p>\n\n\n\nStep 4:<\/strong> Now use the power rule of differentiation d\/dx [f(x)] n<\/sup> = n[f(x)] n-1<\/sup> where n = 1, 2.<\/p>\n\n\n\nd\/dx [13x2<\/sup> – 14x – 4cos(x) – 54] = 13 [2x2-1<\/sup>] – 14 [1x1-1<\/sup>] – 4d\/dx [cos(x)] <\/p>\n\n\n\n = (13 * 2) [x2-1<\/sup>] \u2013 (14 * 1) [x1-1<\/sup>] – 4d\/dx [cos(x)] <\/p>\n\n\n\n = 26 x1<\/sup> \u2013 14 x0<\/sup> – 4d\/dx [cos(x)] <\/p>\n\n\n\n = 26x \u2013 14(1) – 4d\/dx [cos(x)] <\/p>\n\n\n\n
= 26x \u2013 14 – 4d\/dx [cos(x)] <\/p>\n\n\n\n
Step 5:<\/strong> Use the chain rule to find the differential of cos(x).<\/p>\n\n\n\nd\/dx cos(x) = d(cos(u))\/du * du\/dx, where u = x<\/p>\n\n\n\n
d\/dx [13x2<\/sup> – 14x – 4cos(x) – 54] = 26x \u2013 14 \u2013 4(-sin(x)) [d(x)\/dx] <\/p>\n\n\n\n = 26x \u2013 14 + 4sin(x) [1] <\/p>\n\n\n\n
= 26x \u2013 14 + 4sin(x)<\/p>\n\n\n\n
= 26x + 4sin(x) \u2013 14<\/p>\n\n\n\n
<\/span>Summary <\/strong><\/span><\/h3>\n\n\n\nIn this article, we\u2019ve learned about the definition, working, and rules of differentiation along with examples. Now you can solve any problem of differential equations just by learning the basics of this post.\u00a0<\/p>\n","protected":false},"excerpt":{"rendered":"
A derivative is a kind of calculus that is used widely to differentiate functions according to their variables. Calculus is a branch of mathematics frequently used to solve complex problems or find changes in functions. This\u00a0branch of mathematics\u00a0allows us to solve problems mathematically because, before calculus, all problems were solved statistically. Derivatives, integrals, limits, and […]<\/p>\n","protected":false},"author":4,"featured_media":11282,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[76],"tags":[1481,1480,170],"class_list":["post-11278","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-statistics","tag-definition-of-derivative","tag-derivative-in-calculus","tag-statistics"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/posts\/11278","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/comments?post=11278"}],"version-history":[{"count":3,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/posts\/11278\/revisions"}],"predecessor-version":[{"id":38288,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/posts\/11278\/revisions\/38288"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/media\/11282"}],"wp:attachment":[{"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/media?parent=11278"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/categories?post=11278"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/statanalytica.com\/blog\/wp-json\/wp\/v2\/tags?post=11278"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}