Introduction to the Derivative of secx tanx
Derivatives have a wide range of applications in almost every field of engineering and science. The secx tanx differentiation can be calculated by following the rules of differentiation. Or, we can directly find the derivative of secxtanx by applying the first principle of differentiation. In this article, you will learn what the derivative of secx tanx is and how to calculate the derivative by using different approaches.
What is the derivative of secxtanx?
The derivative of sec x.tan x, denoted by d/dx(sec x.tan x), is the rate of change of the product of sec x and tan x with respect to the variable x.
The formula for the derivative is sec x times the quantity of sec x squared plus tan x squared. In other words, the secx tanx differentiation is equal to sec x(sec^2x+tan^2x). Knowing the derivative of sec x.tan x is essential in calculus, and it helps in finding the slopes of curves in trigonometric functions.
Derivative of sec x tan x formula
The formula for the derivative of sec x.tan x is equal to sec x times the quantity of sec x squared plus tan x squared, which can be expressed as:
d/dx(sec x.tan x) = sec x(sec^2x+tan^2x)
This formula represents the rate of change of the product of sec x and tan x with respect to the variable x. By understanding this formula, you can solve problems related to calculus and trigonometric functions that involve finding the slopes of curves.
How do you prove the derivative of secx tanx?
There are various differentiation rules to derive the differentiation of secx tanx. Therefore, we can prove the derivative of sec x.tan x by using;
Quotient Rule
Product Rule
Each method provids a different way to compute the sec xtanx differentiation. By using these methods, we can mathematically prove the formula for finding differential of sec xtan x.
Differentiation of sec x tan x using quotient rule
Another method for finding the derivative of sec xtan x is using the quotient rule, which is a formula for finding the derivative of a quotient of two functions. The derivative of cosecant can also be calculated using the quotient rule. The quotient rule is defined as:
d / dx (f/g) = f(x). g'(x) -g(x).f'(x) /{g(x)}2
Proof of derivative of secxtanx by quotient rule
To prove the derivative of sec x tan x, we can start by writing it as,
f(x) = sec xtan x = sin x/cos2x =u/v
Supposing that u = sin x and v = cos2 x. Now by quotient rule,
f'(x) = (vu' - uv')/v2
f(x) = [cos2x. d / dx(sin x) - sin x.d / dx(cos2 x)] / (cos2 x)2
= [-cos3x + sin x (-2sin x.cos x)] / cos4x
= sec x(sec2x+tan2x)
Hence, we have derived the differentiation of sec x tan x using the quotient rule of differentiation. We can also calculate the derivative of sec x by using quotient rule.
Derivative of secx tanx by product rule
The secx tanx derivative can be calculated by using product rule because an algebraic function can be written as the combination of two functions. The product rule of derivatives is defined as;
[uv]' = u.v' +u'.v
Proof of secxtanx derivative by product rule
To prove the differentiation of secx tanx by using product rule calculator, assume that,
f(x) = sec x.tan x
By using product rule of differentiation,
f'(x) = sec x.(tan x)'+tan x. (sec x)'
We get,
f'(x) = sec x.sec2x + tan x(sec x tan x)
f'(x) = sec3x+tan2x sec x
Hence,
f'(x) = sec x(sec2x+tan2x)
It is the secx tanx derivative calculated by using the product rule formula.
How to find the secxtanx derivative with a calculator?
The easiest way to differentiate secx tanx is by using an online tool derivative finder. You can use our derivative calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.
Write the function as sec x.tan x in the enter function box. In this step, you need to provide input value as a function as you have to differentiate secx tanx.
Now, select the variable by which you want to differentiate sec x.tan x. Here you have to choose 'x'.
Select how many times you want to differentiate sec x.tan x. In this step, you can choose 2 for second, 3 for third derivative and so on.
Click on the calculate button.
After completing these steps, you will receive the differentiation of secx tanx within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.
Frequently Asked Questions
What is the second derivative of secx?
The first derivative of sec x is equal to sec x.tan x. The second derivative of sec x can be calculated by applying differentiation on the first derivative of sec x.
d / dx (sec x) = sec x.tan x
d2 / dx22 (sec x) = sec x.(tan x)' + tan x.(sec x)'
d2 / dx22 (sec x) = sec x(sec2x+tan2x)
What is secx tanx?
In mathematics, sec x and tan x are the trigonometric identities. Sec x is equal to the reciprocal of 1 and cos x and tan x is ratio of sin x and cos x. In derivative, sec x.tan x is the derivative of sec x.
What is the derivative of tangent?
The derivative of tangent is secant squared and the derivative of cotangent is negative cosecant squared.