Derivative of Secx
Sec(x) Derivative Rule
Secant is the reciprocal of the cosine. The secant of an angle designated by a variable x is notated as sec(x).
The derivative rule for sec(x) is given as:
let’s find the derivative…
of the secant of x.
How would we go about doing this?
Well sec(x) is another way of writing 1/cos(x)…
and since we have a quotient here,
the quotient rule would aptly apply…
so written in short hand, the quotient rule is:
y-dash or y-prime… I’m going to call it y-dash…
is equal to (vu’ – uv’) / v^2.And in this case, we have y = sec(x), and…
I’m going to let u equal the numerator of 1
and v equal the denominator: cos(x).
So now if we apply the quotient rule…
for the first part, we have v, so cos(x) remains unchanged…
multiplied by the derivative of u.
So the derivative of 1, a constant, is zero…
then minus u, which is one, times the derivative of v…
which is the derivative of cos(x)…
and that’s equal to -sin(x)…
all over v^2, which is cos^2(x).
So the result is positive sin(x)…
divided by the cos^2(x).
And I can rewrite this as…
[sin(x)/cos(x)]*1/cos(x).
And then I can realise that this part is equal to…
the tangent of x… and this part is equal to again…
the secant of x!
So the derivative of sec(x) is equal to…
tan(x)*sec(x)
Derivative of sec(x)
Sec(x) Derivative Rule
Secant is the reciprocal of the cosine. The secant of an angle designated by a variable x is notated as sec(x).
The derivative rule for sec(x) is given as:
^{d}⁄_{dx}sec(x) = tan(x)sec(x)
This derivative rule gives us the ability to quickly and directly differentiate sec(x).
X may be substituted for any other variable.
For example, the derivative ^{d}⁄_{dy}sec(y) = tan(y)sec(y), and the derivative ^{d}⁄_{dz}sec(z) = tan(z)sec(z).
Proof of the Derivative Rule
The sec(x) derivative rule is originates from the relation that sec(x) = 1/cos(x). Now, the first step of finding the derivative of 1/cos(x) is using the quotient rule.
Using the quotient rule on 1/cos(x) gives us:
[sin(x)/cos(x)][1/cos(x)]
sin(x)/cos(x) = tan(x), and 1/cos(x) = sec(x)
Therefore, it simplifies to tan(x)sec(x), resulting in:
^{d}⁄_{dx}sec(x) = tan(x)sec(x)
Derivative Rules of the other Trigonometry Functions
Derivative Rules Here’s the derivative rules for the other five major trig functions:
^{d}⁄_{dx}sin(x) = cos(x)
^{d}⁄_{dx}cos(x) = -sin(x)
^{d}⁄_{dx}tan(x) = sec^{2}(x)
^{d}⁄_{dx}cot(x) = -csc^{2}(x)
^{d}⁄_{dx}csc(x) = -cot(x)csc(x)
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Antiderivative of secxtanx
Antiderivative of secxtanx
We have to find the antiderivative of secx tanx. To find the antiderivative of secx
tanx we must first understand the meaning or definition of antiderivative.
Antiderivative is an operation which is opposite of derivation operation that
means antiderivative calculates the integral of a derivative.
Suppose we have a function f(x), its derivative is g(x) means d(f(x))= g(x) than
antiderivative of g(x) is f(x) that is J g(x) dx = f(x) + c dx. So we get f(x) as
antiderivative of g(x) is a constant.
We have understood the meaning of antiderivative, so now we will calculate the
antiderivative of secx tanx.
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The process of finding antiderivative of sec(x) tan(x) is shown in steps below-
Step 1 : Find the derivative of sec(x), that is d(sec x) / dx = sec(x) tan(x) dx.
Therefore g(x) sec(x) tan(x).
Step 2 : Now to find the antiderivative of sec(x) tan(x) , we will have to find the
integral of sec(x) tan(x).
That is Jsec(x) tan(x) dx = sec(x).
Proof of above operation- => Let g(x)= sec x tan x dx => g(x) = J sin x / cos 2 x
dx ( sec x = I / cos x , tan x = sin x / cos x) Let u = cosx
Therefore du – sin x dx g(x) – f 1 / u 2 du g(x) – u-2 du g(x) = 1/
u + C => g(x) = 1/ cos x + C => g(x) sec x + C Hence proved. Here we have
used method of integration by parts to calculate the antiderivative of sec x tan x.
Hence to find the antiderivative of secxtanx we have to calculate the integral of