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

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

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Find Derivative with Chain Rule: We need to find the derivative of the function with respect to v. We will use the chain rule for the term involving u, since u is a function of v. 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.Differentiate 5v^(5): First, let's differentiate the term 5v^(5) with respect to v. The power rule for derivatives tells us that d/dv(v^n) = n*v^(n-1), so the derivative of 5v^(5) with respect to v is 5 * 5v^(4) = 25v^(4).Differentiate -4sin(u): Now, let's differentiate the term -4sin(u) with respect to v. Since u is a function of v, we need to use the chain rule. The derivative of -4sin(u) with respect to u is -4cos(u), and we need to multiply this by the derivative of u with respect to v, which is d/dv(4v^(2) + 1).Apply Chain Rule for u: The derivative of u with respect to v is d/dv(4v^(2) + 1) = 8v, because the derivative of a constant is 0 and the derivative of 4v^(2) is 8v using the power rule.Combine Derivatives: Now, we multiply the derivative of -4sin(u) with respect to u by the derivative of u with respect to v to get the full derivative of -4sin(u) with respect to v. This gives us -4cos(u) * 8v = -32v*cos(u).Express in Terms of v: Finally, we combine the derivatives of both terms to get the full derivative of the function with respect to v. The derivative of 5v^(5) - 4sin(u) with respect to v is 25v^(4) - 32v*cos(u).We need to express the derivative in terms of v only, so we substitute u = 4v^(2) + 1 into the cosine term. This gives us -32v*cos(4v^(2) + 1).

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

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Given that u=4v2+1 u=4 v^{2}+1 u=4v2+1, find ddv(5v5−4sin⁡u) \frac{d}{d v}\left(5 v^{5}-4 \sin u\right) dvd​(5v5−4sinu) in terms of only v v v.\newlineAnswer:

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