Question

Differentiate cosx/sin2x​

Answer 1

derivative of cosx/sin2x is (1 + secx) / [2sinx cosx].

Explanation:

To differentiate cosx/sin2x, we can use the quotient rule of differentiation, which states:

(d/dx) [f(x)/g(x)] = [g(x) * f'(x) - f(x) * g'(x)] / [g(x)]^2

Here, f(x) = cosx and g(x) = sin2x. So we have:

f'(x) = -sinx (differentiation of cosine)

g'(x) = 2cos2x (differentiation of sine2x using chain rule)

Now, using the quotient rule:

(d/dx) [cosx/sin2x] = [sin2x * (-sinx) - cosx * (2cos2x)] / [sin2x]^2

= [-2cos2x cosx + sin2x sinx] / [sin2x]^2

= [-2cosx cos2x + sinx] / [2sinx cosx]^2

= [-2cosx(1-2sin^2(x)) + sinx] / [2sinx cosx]^2

= [-2cosx + 4cosx sin^2(x) + sinx] / [2sinx cosx]^2

= (sinx + 2cosx sin^2(x)) / [2sinx cosx]^2

= sinx / [2sinx cosx]^2 + 2cosx / [2sinx cosx]^2

= 1/[2sinx cosx] + secx/[2sinx cosx]

= (1 + secx) / [2sinx cosx]

Therefore, the derivative of cosx/sin2x is (1 + secx) / [2sinx cosx].