Given that x=3v^(3)+2, find (d)/(dv)(4v^(2)+5cos x) in terms of only v. Answer:

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Given that  x=3v^(3)+2, find  (d)/(dv)(4v^(2)+5cos x) in terms of only  v. Answer:

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Find Derivative: First, we need to find the derivative of the function 4v^2 + 5cos(x) with respect to v. To do this, we will use the chain rule, which 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. The outer function in this case is 4v^2 + 5cos(u), where u is a function of v, and the inner function is u(x) = x = 3v^3 + 2.Apply Chain Rule: We start by differentiating the outer function with respect to v, treating x as a constant for now. The derivative of 4v^2 with respect to v is 8v. Since x is treated as a constant, the derivative of 5cos(x) with respect to v is 0. So the derivative of the outer function with respect to v is 8v.Differentiate Inner Function: Next, we need to differentiate the inner function x = 3v^3 + 2 with respect to v. The derivative of 3v^3 with respect to v is 9v^2, and the derivative of the constant 2 with respect to v is 0. So the derivative of the inner function with respect to v is 9v^2.Apply Chain Rule Again: Now we apply the chain rule. We multiply the derivative of the outer function by the derivative of the inner function. The derivative of the outer function is 8v, and the derivative of the inner function is 9v^2. Therefore, the derivative of 4v^2 with respect to v is 8v * 9v^2 = 72v^3.Combine Derivatives: However, we also need to differentiate the term 5cos(x) with respect to v using the chain rule. The derivative of cos(x) with respect to x is -sin(x), and we already found that the derivative of x with respect to v is 9v^2. So the derivative of 5cos(x) with respect to v is -5sin(x) * 9v^2.Express in Terms of v: Now we combine the derivatives of both terms to get the full derivative of the function with respect to v. The derivative of 4v^2 is 72v^3, and the derivative of 5cos(x) is -5sin(x) * 9v^2. So the full derivative is 72v^3 - 45v^2sin(x).Final Derivative: Finally, we need to express the derivative in terms of v only. Since x = 3v^3 + 2, we substitute this into the sine function to get sin(3v^3 + 2). The final derivative of the function with respect to v is 72v^3 - 45v^2sin(3v^3 + 2).

Given that $x=3 v^{3}+2$ , find $\frac{d}{d v}\left(4 v^{2}+5 \cos x\right)$ in terms of only $v$ . $\newline$ Answer:

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Given that x=3v3+2 x=3 v^{3}+2 x=3v3+2, find ddv(4v2+5cos⁡x) \frac{d}{d v}\left(4 v^{2}+5 \cos x\right) dvd​(4v2+5cosx) in terms of only v v v.\newlineAnswer:

Given that $x=3 v^{3}+2$ , find $\frac{d}{d v}\left(4 v^{2}+5 \cos x\right)$ in terms of only $v$ . $\newline$ Answer: