{"id":33979,"date":"2024-01-10T10:46:46","date_gmt":"2024-01-10T10:46:46","guid":{"rendered":"https:\/\/www.greatassignmenthelp.com\/questions\/?p=33979"},"modified":"2024-01-12T10:09:11","modified_gmt":"2024-01-12T10:09:11","slug":"derivative-of-sec-x","status":"publish","type":"post","link":"https:\/\/www.greatassignmenthelp.com\/questions\/derivative-of-sec-x\/","title":{"rendered":"What Is the Derivative of Sec x"},"content":{"rendered":"

Answer:<\/strong> The trigonometric function sec x is the reciprocal of cos x. The derivative of the sine function is expressed as sin\u2032(a)= cos(a), which indicates that the cosine of an angle is what determines the rate of change of sin x at a particular angle, x= an.<\/p>\n

Formula for Derivative of Sec x<\/strong><\/h2>\n

The derivative or differentiation of a secant function with respect to a variable is equal to the product of the tangent and secant functions. Tan x is the formula for the derivative, and it states that the derivative of the sec x function with respect to x is equal to the product of sec x.<\/p>\n

The secant function is expressed in mathematical notation using trigonometry, assuming that x is a variable.<\/p>\n

d\/dx(secx) = secxtanx<\/strong><\/h3>\n

In mathematics, the differentiation of a sec x function for\u00a0x can be written as (sec x)\u2019 or as d\/dx (sec x).<\/p>\n

However, you can express the derivative of the secant function in terms of any variable such as:<\/p>\n

    \n
  • d\/dh (sec h) = sec h\u00a0tan h<\/li>\n
  • d\/dw (sec w) = sec w tan w<\/li>\n
  • d\/dy (sec y) = sec y tan y<\/li>\n<\/ul>\n

    Proof of the Sec x Derivative<\/strong><\/h2>\n

    For a given variable, the derivative of a secant function is equal to the product of the tangent and secant functions. The mathematical expression of the secant function, when \u2018x\u2019 is utilized as a variable, is x\u2019. The differentiation between \u201csec x\u201d and \u201cx\u201d is the product of \u201csec x\u201d and \u201ctan x.\u201d The first principle of differentiation in differential calculus is used to get the derivative of a secant function.<\/p>\n

    Derivative of Sec x<\/strong><\/h3>\n

    Let\u2019s review a few things before attempting to get the derivative of sec x. The ratio of sin x to cos x is known as tan x, and the reciprocal of cos x is known as sec x. It is crucial to differentiate sec x from\u00a0x using sec x and tan x definitions. There are several ways we can find it, including:<\/p>\n

      \n
    • utilizing the first principle<\/li>\n
    • by applying the rule of quotient<\/li>\n
    • use the chain rule<\/li>\n<\/ul>\n

      Let\u2019s differentiate sec x using each of these techniques, and then use the derivative of sec x to answer a few problems.<\/p>\n

      Derivative of Sec x by First Principle<\/strong><\/h2>\n

      Using fundamental principles (or the definition of the derivative), we will demonstrate that sec x\u2019s derivative is sec x \u00b7 tan x. Now, let\u2019s assume assume f(x) = sec x<\/p>\n

      Proof: <\/strong>The derivative of a function f(x) is,<\/p>\n

      f\u2019(x) = lim\u2095\u2192\u2080 [f (x + h) \u2013 f(x)] \/ h \u2026 (1)<\/p>\n

      Since f(x) = sec x, we have f (x + h) = sec (x + h)<\/p>\n

      When these values are substituted in (1),<\/p>\n

      f\u2019 (x) = lim\u2095\u2192\u2080 [sec (x + h) \u2013 sec x]\/h<\/p>\n

      = lim\u2095\u2192\u2080 1\/h [1\/ (cos (x + h) \u2013 1\/cos x)]<\/p>\n

      = lim\u2095\u2192\u2080 1\/h [cos x \u2013 cos (x + h)] \/ [cos x cos (x + h)]<\/p>\n

      For cos A \u2013 cos B, the sum to product formulas provides cos (A+B)\/2 sin (A-B)\/2 = -2.<\/p>\n

      The formula for f\u2019(x) is 1\/cos x lim\u2095\u2192\u2080 1\/h [- 2 sin (x + x + h)\/2 sin (x \u2013 x \u2013 h)\/2]. \/ [cos (h + x)]<\/p>\n

      = 1\/cos x lim\u2095\u2192\u2080 1\/h [- 2 sin (2x + h)\/2 sin (- h)\/2] \/ [cos (x + h)]<\/p>\n

      Divide and multiply by h\/2,<\/p>\n

      = 1\/cos x lim\u2095\u2192\u2080 (1\/h) (h\/2) [- 2 sin (2x + h)\/2 sin (- h\/2) \/ (h\/2)] \/ [cos (x + h)]<\/p>\n

      Sin (h\/2) \/ (h\/2) = 1\/cos x lim\u208e\/\u2082\u2192\u2080 f\u2019(x). cos (x + h)\/sin (2x + h) = lim\u208e\u2192\u2080<\/p>\n

      Given lim\u2093\u2192\u2080 (sin x) \/ x = 1, we get,<\/p>\n

      sin x\/cos x = f\u2019(x) = 1\/cos x<\/p>\n

      We are aware that sin x\/cos x = tan x and 1\/cos x = sec x. Thus,<\/p>\n

      Sec x \u00b7 tan x = f\u2019(x)<\/p>\n

      Sec x Derivative using Quotient Rule<\/strong><\/h3>\n

      Using the quotient rule, we will demonstrate that sec x \u00b7 tan x is the result of differentiating sec x about\u00a0x. To\u00a0do this, we will suppose that f(x) = sec x, or f(x) = 1\/cos x.<\/p>\n

      Proof: <\/strong>The formula is f(x) = 1\/cos x = u\/v.<\/p>\n

      By using the quotient rule,<\/p>\n

      If vu\u2019 \u2013 uv\u2019 = f\u2019(x) \/ v2, then<\/p>\n

      f\u2019(x) = [cos x d\/dx (1) \u2013 1 d\/dx (cos x)] \/ (cos x)2<\/p>\n

      [cos x (0) \u2013 1 (-sin x)] \/ cos2x<\/p>\n

      (sin x) \/ cos2x<\/p>\n

      1\/cos x \u00b7 (sin x)\/ (cos x)<\/p>\n

      sec x \u00b7 tan x<\/p>\n

      Chain Rule Derivative of Sec x<\/strong><\/h3>\n

      Since f(x) = sec x = 1\/cos x, we will determine that the derivative of sec x is sec x \u00b7 tan x by chain rule.<\/p>\n

      Proof:<\/strong> f(x) can be expressed as,<\/p>\n

      f(x) = 1\/cos x = (cos x)-1<\/p>\n

      By the rules of chain and power,<\/p>\n

      f\u2019(x) = (-1) (cos x)-2 d\/dx (cos x)<\/p>\n

      This means that a-m = 1\/am by an exponent characteristic. We are also aware that -sin x = d\/dx (cos x). So,<\/p>\n

      f\u2019(x) = -1\/cos2x \u00b7 (- sin x)<\/p>\n

      (sin x) \/ cos2x<\/p>\n

      1\/cos x \u00b7 (sin x)\/ (cos x)<\/p>\n

      sec x \u00b7 tan x<\/p>\n

      Solved Examples of Derivative of sec x<\/strong><\/h3>\n

      Example 1: What is the derivative of (secx)2?<\/strong><\/p>\n

      Solution: <\/strong>f(x)=(secx)2<\/p>\n

      Using the chain rule and the power rule<\/p>\n

      f\u2032(x)=2secxddx(secx)<\/p>\n

      =2secx. (secx.tanx)<\/p>\n

      =2sec2xtanx<\/p>\n

      Here is the function\u2019s derivative: =2sec2xtanx.<\/strong><\/h2>\n

      You can also go for online assignment help services to get solutions and assistance for your derivative of sec x assignment problems.<\/p>\n","protected":false},"excerpt":{"rendered":"

      Answer: The trigonometric function sec x is the reciprocal of cos x. The derivative of the sine function is expressed as sin\u2032(a)= cos(a), which indicates […]<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"yoast_head":"\nWhat Is the Derivative of Sec x<\/title>\n<meta name=\"description\" content=\"The trigonometric function sec x is the reciprocal of cos x. 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