find the derivative of
y= 3-secx/tanx
y' = (3-secx)'tanx -(3-secx)(tanx)'/(tanx)^2
= -secx(tanx)^2 - (3-secx)secx/(tanx)^2
= -secx[(tanx)^2 + (3-secx)]/tan^2x
=-[(tan)^2 +3 -secx]/sinx
Y=-cosxln(secx+tanx)
To find the derivative of y = 3 - sec(x)/tan(x), we can use the quotient rule.
The quotient rule states that if we have a function in the form of f(x) = g(x)/h(x), where g(x) and h(x) are both differentiable functions, then the derivative of f(x) is given by:
f'(x) = [h(x)g'(x) - g(x)h'(x)] / [h(x)]^2
In our case, g(x) = 3 - sec(x) and h(x) = tan(x).
Now, let's differentiate g(x) and h(x) individually:
g'(x) = 0 - sec(x)tan(x) = -sec(x)tan(x)
h'(x) = sec^2(x)
Substituting these values into the quotient rule:
f'(x) = [tan(x)(-sec(x)tan(x)) - (3 - sec(x))(sec^2(x))] / [tan(x)]^2
Simplifying further:
f'(x) = [-sec^2(x)tan^2(x) + (3 - sec(x))(sec^2(x))] / tan^2(x)
Combining like terms:
f'(x) = [-sec^2(x)tan^2(x) + 3sec^2(x) - sec^3(x)] / tan^2(x)
Now, the derivative of y = 3 - sec(x)/tan(x) is given by f'(x) = [-sec^2(x)tan^2(x) + 3sec^2(x) - sec^3(x)] / tan^2(x).
To find the derivative of y = 3 - sec(x)/tan(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 two differentiable functions, then the derivative of f(x) is given by:
f'(x) = (v(x) * u'(x) - u(x) * v'(x)) / [v(x)]^2
Now let's apply the quotient rule to find the derivative of y = 3 - sec(x)/tan(x):
We can rewrite y as y = 3 - (sec(x) / tan(x)) = 3 - sec(x) * (1/tan(x))
Let u(x) = 3 and v(x) = sec(x)*tan(x)
Now, let's find the derivatives of u(x) and v(x):
u'(x) = 0 (since the derivative of a constant is zero)
To find v'(x), we can use the product rule:
v'(x) = [sec(x)]' * [tan(x)] + sec(x) * [tan(x)]'
Using the rules for derivatives, we have:
v'(x) = sec(x) * tan(x) + sec(x)^2
Now, we can substitute the values into the quotient rule formula to find the derivative of y:
y' = (v(x) * u'(x) - u(x) * v'(x)) / [v(x)]^2
= (sec(x) * tan(x) * 0 - 3 * (sec(x) * tan(x) + sec(x)^2)) / [sec(x) * tan(x)]^2
Simplifying further:
y' = (-3 * sec(x) * tan(x) - 3 * sec(x)^2) / [sec(x) * tan(x)]^2
= -3 * (sec(x) * tan(x) + sec(x)^2) / [sec(x) * tan(x)]^2
Finally, we can simplify the expression by dividing every term by [sec(x)]^2:
y' = -3 * (sec(x) * tan(x) + sec(x)^2) / [sec(x) * tan(x)]^2
= -3 * (tan(x) + sec(x)) / [tan(x)]^2
So the derivative of y = 3 - sec(x)/tan(x) is given by y' = -3 * (tan(x) + sec(x)) / [tan(x)]^2.