f(x)=12x/sinx+cosx
find f'(-pie)
anyone know the answer!!! need help!
http://www.jiskha.com/display.cgi?id=1298613079
after you find the derivative, plug in -pie and solve
To find f'(-π), we need to find the derivative of the given function f(x) and evaluate it at x = -π.
Let's start by differentiating f(x) using the quotient rule:
f(x) = (12x) / (sin(x) + cos(x))
Using the quotient rule, we have:
f'(x) = (12(sin(x) + cos(x)) - 12x(cos(x) - sin(x))) / (sin(x) + cos(x))^2
Now, we evaluate f'(-π) by substituting x = -π:
f'(-π) = (12(sin(-π) + cos(-π)) - 12(-π)(cos(-π) - sin(-π))) / (sin(-π) + cos(-π))^2
Since sin(-π) = 0, cos(-π) = -1, and (sin(-π) + cos(-π)) = -1, we can simplify the equation further:
f'(-π) = (12(0 + (-1)) - 12(-π)((-1) - 0)) / (-1)^2
= (-12π) / 1
= -12π
Therefore, f'(-π) = -12π.
To find the derivative of the function f(x), you can use the quotient rule. The quotient rule states that if you have a function in the form f(x) = g(x)/h(x), where g(x) and h(x) are both differentiable functions, then the derivative of f(x) can be found using the formula:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / [h(x)]^2
In this case, g(x) = 12x and h(x) = sin(x) + cos(x). Taking the derivatives of g(x) and h(x), we have:
g'(x) = 12
h'(x) = cos(x) - sin(x)
Now, substitute these values into the quotient rule formula:
f'(x) = (12 * (sin(x) + cos(x)) - 12x * (cos(x) - sin(x))) / [(sin(x) + cos(x))]^2
To find f'(-π), substitute x = -π into the derivative formula:
f'(-π) = (12 * (sin(-π) + cos(-π)) - 12(-π) * (cos(-π) - sin(-π))) / [(sin(-π) + cos(-π))]^2
Simplifying the expression will give you the value of f'(-π).