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

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

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Express y in terms of u: First, we need to express y in terms of u, which is already given as y = 3u^5 + 2. We will use this to find the derivative of y with respect to u (dy/du).Calculate dy/du: Calculate the derivative of y with respect to u (dy/du). Since y = 3u^5 + 2, the derivative is dy/du = d/du(3u^5 + 2) = 15u^4.Find derivative with chain rule: Now, we need to find the derivative of the given expression 2y^3 - 2sin(u) with respect to u. We will use the chain rule for the term 2y^3 and the basic derivative rule for the term -2sin(u).Apply chain rule to 2y^3: Apply the chain rule to the term 2y^3. 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 f(y) = 2y^3 and the inner function is y(u) = 3u^5 + 2.Calculate derivative of -2sin(u): The derivative of the outer function f(y) = 2y^3 with respect to y is f'(y) = 6y^2. We will then multiply this by the derivative of the inner function dy/du, which we found to be 15u^4.Combine derivatives: The derivative of the term 2y^3 with respect to u is therefore 6y^2 * dy/du = 6y^2 * 15u^4. We substitute y = 3u^5 + 2 into this expression to get 6(3u^5 + 2)^2 * 15u^4.Simplify final expression: Now, calculate the derivative of the term -2sin(u) with respect to u. The derivative of -2sin(u) with respect to u is -2cos(u).Combine the derivatives of both terms to get the derivative of the entire expression with respect to u. This gives us d/du(2y^3 - 2sin(u)) = 6(3u^5 + 2)^2 * 15u^4 - 2cos(u).Simplify the expression by distributing and combining like terms. The final derivative in terms of u is 90u^4(3u^5 + 2)^2 - 2cos(u).

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

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