Derivative secx
The derivative of sec(x), denoted as d/dx(sec(x)), can be found using the quotient rule and the properties of trigonometric functions
The derivative of sec(x), denoted as d/dx(sec(x)), can be found using the quotient rule and the properties of trigonometric functions.
To start, let’s express sec(x) in terms of sine and cosine. The reciprocal identity states that sec(x) is equal to 1/cos(x). Therefore, we have:
sec(x) = 1/cos(x)
To find the derivative, we need to differentiate both the numerator and denominator separately. The derivative of 1 is 0, since it is a constant. The derivative of cos(x) can be found using the chain rule, which states that if we have a composition of functions, we need to differentiate the outer function and then multiply it by the derivative of the inner function.
Differentiating cos(x) gives us -sin(x). Now, let’s apply the quotient rule, which states that for a function f(x) = g(x)/h(x), the derivative is given by:
f'(x) = [h(x)g'(x) – g(x)h'(x)] / [h(x)]^2
Applying the quotient rule to our expression:
d/dx(sec(x)) = [cos(x)(0) – (1)(-sin(x))] / [cos(x)]^2
= sin(x) / [cos(x)]^2
Simplifying the expression, we can rewrite it using the identity for tan(x), which is sin(x)/cos(x):
d/dx(sec(x)) = sin(x) / [cos(x)]^2
= sin(x) / cos^2(x)
= tan(x) / cos(x)
= tan(x) * sec(x)
Therefore, the derivative of sec(x) is tan(x) times sec(x), or d/dx(sec(x)) = tan(x) * sec(x).
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