Evaluating the gradient of a function

Please see the attachment..

Find the gradient of the function: (3√θ^3) / 2sin2θ

The gradient is a first-order differential operator that maps functions to vector fields. It is a generalization of the ordinary derivative.

Here we have a function of θ. The gradient is found by differentiating this with respect to θ

grad ((3√θ^3) / 2sin2θ) = d/d θ of [(3θ3/2) / 2sin2θ]

Use the product rule or quotient rule here.

Product rule:
= (3θ3/2) * derivative of (1/2sin2θ) + (1/2sin2θ) * derivative of (3θ3/2)

= (3θ3/2) * derivative of (1/2sin2θ) + (1/2sin2θ) * 3* 3/2 * θ3/2 -1)

= (3θ3/2) * derivative of (1/2sin2θ) + (9 θ1/2/4sin2θ)

= (3θ3/2) * (1/2) *derivative of cosec(2θ) + (9 θ1/2/4sin2θ)

= (3θ3/2) * (1/2) * [-2 Cot(2θ) * Cosec (2θ)] + (9 θ1/2/4sin2θ)

= - (3θ3/2) * Cot(2θ) * Cosec (2θ) + (9 θ1/2/4sin2θ)

= - (3θ3/2) * Cot(2θ) * Cosec (2θ) + (9/4) Cosec2θ * θ1/2

= (9/4) Cosec2θ * θ1/2 - (3θ3/2) * Cot(2θ) * Cosec (2θ)

= [(9/4) θ1/2 - (3θ3/2) * Cot(2θ)] * Cosec (2θ) ---Answer

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