The differentiation of (\\sin(x)) with respect to (x) is a fundamental concept in calculus. The derivative of (\\sin(x)) is given by:<\/p>\n\n\n\n
\\[
\\frac{d}{dx}(\\sin(x)) = \\cos(x).
\\]<\/p>\n\n\n\n
This result is derived from the limit definition of the derivative and is a key part of the differentiation rules for trigonometric functions. The cosine function, $\\cos(x)$, represents the rate of change of $\\sin(x)$ with respect to (x).<\/p>\n\n\n\n
We can proof using the below two formulas<\/p>\n\n\n\n
$\\frac {d}{dx} f(x) =\\displaystyle \\lim_{h \\to 0} \\frac {f (x+h) – f(x)}{h}$<\/p>\n\n\n\n
$\\displaystyle \\lim_{x \\to 0} \\frac {sinx}{x}=1$<\/p>\n\n\n\n
Now
$\\frac {d}{dx} (sin x) = \\displaystyle \\lim_{h \\to 0} \\frac {sin (x+h) – sin x}{h}= \\displaystyle \\lim_{h \\to 0} \\frac {2cos (\\frac {2x+h}{2}) sin \\frac {h}{2}}{h}$
$=\\displaystyle \\lim_{h \\to 0} cos (x + \\frac {h}{2}) \\displaystyle \\lim_{h \\to 0} \\frac {sin \\frac {h}{2}}{\\frac {h}{2}} = cos x$
as $\\displaystyle \\lim_{x \\to 0} \\frac {sinx}{x}=1$<\/p>\n\n\n\n
\\[
\\frac{d}{dx}(\\sin(x)) = \\cos(x).
\\]<\/p>\n\n\n\n
We can clearly see that Domain of the Derivative of sin x is R<\/strong><\/p>\n\n\n\n The geometric meaning of the derivative of $\\sin(x)$, which is $\\cos(x)$, can be understood by examining the graph of $\\sin(x)$ and considering what the derivative represents.<\/p>\n\n\n\n Question 1:<\/strong> Question 2:<\/strong> \\[ Question 3:<\/strong> \\[ The differentiation of (\\sin(x)) with respect to (x) is a fundamental concept in calculus. The derivative of (\\sin(x)) is given by: \\[\\frac{d}{dx}(\\sin(x)) = \\cos(x).\\] This result is derived from the limit definition of the derivative and is a key part of the differentiation rules for trigonometric functions. The cosine function, $\\cos(x)$, represents the rate of […]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","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":"","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":[498],"tags":[],"class_list":["post-8568","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"\nGeometrical Meaning of the Derivative <\/h2>\n\n\n\n
\n
Solved Questions<\/h2>\n\n\n\n
Differentiate $y = \\sin(3x)$.
Solution:<\/strong>
To differentiate $y = \\sin(3x)$, we use the chain rule. Let (u = 3x), then $y = \\sin(u)$. Differentiating (y) with respect to (x):
\\[
\\frac{dy}{dx} = \\frac{dy}{du} \\cdot \\frac{du}{dx} = \\cos(u) \\cdot 3 = 3\\cos(3x).
\\]<\/p>\n\n\n\n
Find the derivative of $y = x \\sin(x)$.
Solution<\/strong>
Here, we use the product rule. If (y = uv) where (u = x) and $v = \\sin(x)$, then (y’ = u’v + uv’). Therefore:<\/p>\n\n\n\n
\\frac{dy}{dx} = 1 \\cdot \\sin(x) + x \\cdot \\cos(x) = \\sin(x) + x\\cos(x).
\\]<\/p>\n\n\n\n
What is the second derivative of $y = \\sin(x)$?
Solution:<\/strong>
First, we find the first derivative, which is $y’ = \\cos(x)$. Then, differentiating again:<\/p>\n\n\n\n
y” = \\frac{d}{dx}(\\cos(x)) = -\\sin(x).
\\]<\/p>\n","protected":false},"excerpt":{"rendered":"