Given that y=2u^(3)+1, find (d)/(du)(4u^(5)-5sin y) in terms of only u. Answer:

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Given that  y=2u^(3)+1, find  (d)/(du)(4u^(5)-5sin y) in terms of only  u. Answer:

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Find Derivative of 4u^(5): First, we need to find the derivative of the function 4u^(5) - 5sin(y) with respect to u. We will use the chain rule for the term involving y, since y is a function of u.Derivative of -5sin(y): The derivative of 4u^(5) with respect to u is 20u^(4).Apply Chain Rule: To find the derivative of -5sin(y) with respect to u, we need to apply 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. In this case, the outer function is -5sin(y) and the inner function is y(u) = 2u^(3) + 1.Derivative of y with respect to u: The derivative of -5sin(y) with respect to y is -5cos(y).Multiply Derivatives: Now we need to find the derivative of y with respect to u, which is the derivative of 2u^(3) + 1 with respect to u.Substitute y into Expression: The derivative of 2u^(3) with respect to u is 6u^(2), and the derivative of a constant (1) is 0. So, the derivative of y with respect to u is 6u^(2).Combine Derivatives: Multiplying the derivative of -5sin(y) with respect to y by the derivative of y with respect to u gives us the derivative of -5sin(y) with respect to u. This is -5cos(y) * 6u^(2).Simplify Expression: Substituting y = 2u^(3) + 1 into -5cos(y) * 6u^(2) gives us -5cos(2u^(3) + 1) * 6u^(2).Get Final Answer: Combining the derivatives of 4u^(5) and -5sin(y) with respect to u, we get the total derivative: 20u^(4) - 5cos(2u^(3) + 1) * 6u^(2).Simplifying the expression, we get the final answer: 20u^(4) - 30u^(2)cos(2u^(3) + 1).

Given that $y=2 u^{3}+1$ , find $\frac{d}{d u}\left(4 u^{5}-5 \sin y\right)$ in terms of only $u$ . $\newline$ Answer:

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Given that $y=2 u^{3}+1$ , find $\frac{d}{d u}\left(4 u^{5}-5 \sin y\right)$ in terms of only $u$ . $\newline$ Answer: