Given that x=w^(5)+5, find (d)/(dw)(3x^(2)+3cos w) in terms of only w. Answer:

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Given that  x=w^(5)+5, find  (d)/(dw)(3x^(2)+3cos w) in terms of only  w. Answer:

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Express x in terms of w: First, we need to express x in terms of w to be able to differentiate the function with respect to w. We are given that x = w^5 + 5. We will use this to differentiate the function 3x^2 + 3cos(w) with respect to w.Apply chain rule: Next, we need to apply the chain rule to differentiate 3x^2 with respect to w. 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 3u^2 (where u = x) and the inner function is x = w^5 + 5.Derivative of x with respect to w: The derivative of 3u^2 with respect to u is 6u. So, the derivative of 3x^2 with respect to x is 6x. Now we need to multiply this by the derivative of x with respect to w, which is the derivative of w^5 + 5 with respect to w.Multiply derivatives: The derivative of w^5 with respect to w is 5w^4, and the derivative of a constant (5) is 0. Therefore, the derivative of x with respect to w is 5w^4.Substitute x into expression: Now we can multiply the derivative of 3x^2 with respect to x by the derivative of x with respect to w to get the derivative of 3x^2 with respect to w. This gives us 6x * 5w^4.Differentiate 3cos(w): Substituting x = w^5 + 5 into 6x * 5w^4, we get 6(w^5 + 5) * 5w^4. This simplifies to 30w^4(w^5 + 5).Combine derivatives: Now we need to differentiate 3cos(w) with respect to w. The derivative of cos(w) with respect to w is -sin(w). Therefore, the derivative of 3cos(w) with respect to w is -3sin(w).Finally, we add the derivatives of 3x^2 and 3cos(w) with respect to w to get the derivative of the entire function with respect to w. This gives us 30w^4(w^5 + 5) - 3sin(w).

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

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Given that x=w5+5 x=w^{5}+5 x=w5+5, find ddw(3x2+3cos⁡w) \frac{d}{d w}\left(3 x^{2}+3 \cos w\right) dwd​(3x2+3cosw) in terms of only w w w.\newlineAnswer:

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