Given that y=u^(3)+3, find (d)/(du)(3y^(2)+4sin u) in terms of only u. Answer:

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Given that  y=u^(3)+3, find  (d)/(du)(3y^(2)+4sin u) in terms of only  u. Answer:

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Find Derivative of y: First, we need to find the derivative of y with respect to u, since y is a function of u. The given function is y = u^3 + 3. Using the power rule for differentiation, the derivative of u^3 with respect to u is 3u^2, and the derivative of a constant is 0. So, dy/du = 3u^2.Derivative of 3y^2: Next, we need to find the derivative of 3y^2 with respect to y. Using the power rule again, the derivative of y^2 with respect to y is 2y. So, d(3y^2)/dy = 3 * 2y = 6y.Apply Chain Rule: Now, we apply the chain rule to find the derivative of 3y^2 with respect to u. The chain rule states that d(f(g(u)))/du = f'(g(u)) * g'(u). Here, f(y) = 3y^2 and g(u) = y, so we have d(3y^2)/du = d(3y^2)/dy * dy/du = 6y * 3u^2.Substitute y into Expression: We substitute y = u^3 + 3 into the expression we found in the previous step to express it in terms of u. So, d(3y^2)/du = 6(u^3 + 3) * 3u^2.Simplify Expression: Now, we simplify the expression we found in the previous step. d(3y^2)/du = 6(u^3 + 3) * 3u^2 = 18u^2(u^3 + 3).Derivative of 4sin u: Next, we find the derivative of 4sin u with respect to u. The derivative of sin u with respect to u is cos u. So, d(4sin u)/du = 4cos u.Add Derivatives: Finally, we add the derivatives we found for 3y^2 and 4sin u to find the derivative of the entire expression 3y^2 + 4sin u with respect to u. (d)/(du)(3y^2 + 4sin u) = d(3y^2)/du + d(4sin u)/du = 18u^2(u^3 + 3) + 4cos u.

Given that $y=u^{3}+3$ , find $\frac{d}{d u}\left(3 y^{2}+4 \sin u\right)$ in terms of only $u$ . $\newline$ Answer:

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Given that $y=u^{3}+3$ , find $\frac{d}{d u}\left(3 y^{2}+4 \sin u\right)$ in terms of only $u$ . $\newline$ Answer: