d/dx[secx]
To find the derivative of the function f(x) = sec(x) with respect to x, we can use the quotient rule
To find the derivative of the function f(x) = sec(x) with respect to x, we can use the quotient rule.
The quotient rule states that if we have a function in the form f(x) = u(x)/v(x), where u(x) and v(x) are differentiable functions, then the derivative of f(x) with respect to x is given by:
f'(x) = [v(x) * u'(x) – u(x) * v'(x)] / [v(x)]^2
In our case, u(x) = 1 and v(x) = cos(x).
First, let’s find the derivative of u(x), which is u'(x):
u(x) = 1, so u'(x) = 0 (the derivative of a constant is zero).
Next, let’s find the derivative of v(x), which is v'(x):
v(x) = cos(x), so v'(x) = -sin(x) (the derivative of cos(x) is -sin(x)).
Now, we can substitute these values into the quotient rule:
f'(x) = [cos(x) * 0 – 1 * (-sin(x))] / [cos(x)]^2
Simplifying further:
f'(x) = sin(x) / cos^2(x)
We can simplify this expression by using the identity cos^2(x) = 1 + sin^2(x):
f'(x) = sin(x) / (1 + sin^2(x))
And as sec(x) is defined as 1/cos(x), we can rewrite sin(x) as sec(x) * cos(x):
f'(x) = sec(x) * cos(x) / (1 + sec^2(x) * cos^2(x))
Simplifying further:
f'(x) = sec(x) * cos(x) / (1 + sec^2(x))
Therefore, the derivative of sec(x) with respect to x is sec(x) * cos(x) / (1 + sec^2(x)).
More Answers:
[next_post_link]