If y=3/(sinx+cosx) , find dy/dx
A. 3sinx-3cosx
B. 3/(sinx+cosx)^2
C. -3/(sinx+cosx)^2
D. 3(cosx-sinx)/(sinx+cosx)^2
E. 3(sinx-cosx)/(1+2sinxcosx)
ever heard of the chain rule?
y = 3/u
y' = -3/u^2 u'
Now just plug in u = sinx+cosx
To find dy/dx for the given equation, we need to apply the quotient rule of differentiation.
The quotient rule states that if we have a function f(x) = g(x) / h(x), then its derivative f'(x) can be found using the formula:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
In this case, the function is y = 3 / (sinx + cosx).
To find dy/dx, let's differentiate both the numerator and denominator separately.
1. Differentiating the numerator:
The numerator is a constant value, so its derivative is zero.
2. Differentiating the denominator:
The denominator is (sinx + cosx).
The derivative of sinx is cosx.
The derivative of cosx is -sinx.
Next, we can substitute the derivatives obtained into the quotient rule formula:
dy/dx = (0 * (sinx + cosx) - 3 * (cosx - sinx)) / ((sinx + cosx)^2)
Simplifying further, we get:
dy/dx = -3(cosx - sinx) / (sinx + cosx)^2
Therefore, the correct answer is option D: 3(cosx - sinx) / (sinx + cosx)^2.