Given that x=4w^(2)+4, find (d)/(dw)(x^(4)-5sin w) in terms of only w. Answer:

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Given that  x=4w^(2)+4, find  (d)/(dw)(x^(4)-5sin w) in terms of only  w. Answer:

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Expand x^4: First, we need to express x^4 - 5sin(w) in terms of w using the given expression for x. We have x = 4w^2 + 4. To find x^4, we need to raise the expression for x to the fourth power.Calculate x^4: We calculate x^4 by raising (4w^2 + 4) to the fourth power. This involves expanding the binomial (4w^2 + 4)^4 using the binomial theorem or by multiplying the expression by itself four times. However, since we are ultimately interested in the derivative with respect to w, we can simplify our work by differentiating directly without fully expanding the binomial.Chain rule for x^4: The derivative of x^4 with respect to w can be found 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. In this case, the outer function is u^4 (where u = x) and the inner function is x(w) = 4w^2 + 4.Combine derivatives: Using the chain rule, we differentiate x^4 with respect to x and then multiply by the derivative of x with respect to w. The derivative of x^4 with respect to x is 4x^3. The derivative of x with respect to w is the derivative of 4w^2 + 4, which is 8w.Differentiate -5sin(w): Now we combine the two derivatives to find the derivative of x^4 with respect to w: (d/dw)(x^4) = 4x^3 * (d/dw)(x) = 4x^3 * 8w = 32wx^3.Sum of derivatives: Next, we need to differentiate -5sin(w) with respect to w. The derivative of -5sin(w) with respect to w is -5cos(w).Substitute x into derivative: We now have the derivatives of both terms in the expression x^4 - 5sin(w). The derivative of the entire expression with respect to w is the sum of the derivatives of the individual terms: (d/dw)(x^4 - 5sin(w)) = 32wx^3 - 5cos(w).Finally, we substitute x = 4w^2 + 4 into the expression for the derivative to express it entirely in terms of w: (d/dw)(x^4 - 5sin(w)) = 32w(4w^2 + 4)^3 - 5cos(w).

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

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Given that x=4w2+4 x=4 w^{2}+4 x=4w2+4, find ddw(x4−5sin⁡w) \frac{d}{d w}\left(x^{4}-5 \sin w\right) dwd​(x4−5sinw) in terms of only w w w.\newlineAnswer:

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