Find the derivative of y. y=3t^(3)tanh((1)/(t^(2)))

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Find the derivative of  y.  y=3t^(3)tanh((1)/(t^(2)))

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Apply Product Rule: step_1: Apply the product rule for differentiation, which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Let u = 3t^3 and v = tanh(1/t^2). Then y = u * v. We need to find u' (the derivative of u with respect to t) and v' (the derivative of v with respect to t).Differentiate u: step_2: Differentiate u = 3t^3 with respect to t. u' = d/dt(3t^3) = 3 * d/dt(t^3) = 3 * 3t^2 = 9t^2.Differentiate v: step_3: Differentiate v = tanh(1/t^2) with respect to t using the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. v' = d/dt(tanh(1/t^2)) = sech^2(1/t^2) * d/dt(1/t^2).Differentiate Inner Function: step_4: Differentiate the inner function 1/t^2 with respect to t. Let w = 1/t^2, then w' = d/dt(1/t^2) = d/dt(t^(-2)) = -2t^(-3) = -2/t^3.Combine Results: step_5: Combine the results from step 3 and step 4 to find v'. v' = sech^2(1/t^2) * (-2/t^3).Use Product Rule: step_6: Use the product rule to find the derivative of y = u * v. y' = u' * v + u * v'. Substitute u' = 9t^2 from step 2 and v' from step 5 into the equation. y' = 9t^2 * tanh(1/t^2) + 3t^3 * (sech^2(1/t^2) * (-2/t^3)).Simplify Expression: step_7: Simplify the expression for y'. y' = 9t^2 * tanh(1/t^2) - 6 * sech^2(1/t^2).total_number_of_steps=7

Find the derivative of $y$ . $\newline$ $y=3t^{3}\tanh\left(\frac{1}{t^{2}}\right)$

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Find the derivative of $y$ . $\newline$ $y=3t^{3}\tanh\left(\frac{1}{t^{2}}\right)$