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