d/dx [secx]
To find the derivative of the function sec(x), we need to use the differentiation rules and techniques
To find the derivative of the function sec(x), we need to use the differentiation rules and techniques. In this case, we can start by rewriting sec(x) as 1/cos(x).
Let’s solve it step by step:
Step 1: Rewrite sec(x) as 1/cos(x):
f(x) = 1/cos(x)
Step 2: Apply the Quotient Rule for differentiation:
To differentiate f(x) = u/v, where u = 1 and v = cos(x), we use the Quotient Rule, which states:
(f/g)’ = (g*f’ – f*g’)/g^2
Differentiating both numerator and denominator separately, we get:
f'(x) = (0*cos(x) – 1*(-sin(x)))/(cos(x))^2
= -(-sin(x))/(cos(x))^2
= sin(x)/(cos(x))^2
Step 3: Simplify the expression:
To simplify the expression further, we can rewrite sin(x) as 1/csc(x) and cos(x) as 1/sec(x):
f'(x) = (1/csc(x))/((1/sec(x))^2)
= (1/csc(x))/(1/sec(x))^2
= (1/csc(x))/(1/sec(x))*(1/sec(x))
= (sec(x))/(csc(x))
= sec(x)*sin(x)
Therefore, the derivative of sec(x) is sec(x)*sin(x).
In summary:
d/dx [sec(x)] = sec(x)*sin(x)
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