# Evaluating the gradient of a function

Find the gradient of the function: (3âˆšÎ¸^3) / 2sin2Î¸

I have a number of these questions to complete could you please explain each step involved to get the correct answer

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Find the gradient of the function: (3âˆšÎ¸^3) / 2sin2Î¸

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 Î¸. The gradient is found by differentiating this with respect to Î¸

grad ((3âˆšÎ¸^3) / 2sin2Î¸) = d/d Î¸ of [(3Î¸3/2) / 2sin2Î¸]

Use the product rule or quotient rule here.

Product rule:

= (3Î¸3/2) * derivative of (1/2sin2Î¸) + (1/2sin2Î¸) * derivative of (3Î¸3/2)

= (3Î¸3/2) * derivative of (1/2sin2Î¸) + (1/2sin2Î¸) * 3* 3/2 * Î¸3/2 -1)

= (3Î¸3/2) * derivative of (1/2sin2Î¸) + (9 Î¸1/2/4sin2Î¸)

= (3Î¸3/2) * (1/2) *derivative of cosec(2Î¸) + (9 Î¸1/2/4sin2Î¸)

= (3Î¸3/2) * (1/2) * [-2 Cot(2Î¸) * Cosec (2Î¸)] + (9 Î¸1/2/4sin2Î¸)

= - (3Î¸3/2) * Cot(2Î¸) * Cosec (2Î¸) + (9 Î¸1/2/4sin2Î¸)

= - (3Î¸3/2) * Cot(2Î¸) * Cosec (2Î¸) + (9/4) Cosec2Î¸ * Î¸1/2

= (9/4) Cosec2Î¸ * Î¸1/2 - (3Î¸3/2) * Cot(2Î¸) * Cosec (2Î¸)

= [(9/4) Î¸1/2 - (3Î¸3/2) * Cot(2Î¸)] * Cosec (2Î¸) ---Answer

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