Given that y=2x^(4)+1, find (d)/(dx)(y^(3)+4cos x) in terms of only x. Answer:

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Given that  y=2x^(4)+1, find  (d)/(dx)(y^(3)+4cos x) in terms of only  x. Answer:

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Apply Chain Rule: First, we need to apply the chain rule to find the derivative of y^3 with respect to x. 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(u) = u^3 and the inner function is u = y, where y is a function of x.Derivative of y^3: The derivative of the outer function f(u) = u^3 with respect to u is 3u^2. We will substitute u with y to get 3y^2.Derivative of 4cos(x): Now we need to find the derivative of the inner function y with respect to x. Since y = 2x^4 + 1, the derivative of y with respect to x is dy/dx = d/dx(2x^4 + 1) = 8x^3.Combine Derivatives: Multiplying the derivative of the outer function by the derivative of the inner function, we get the derivative of y^3 with respect to x: (d/dx)(y^3) = 3y^2 * dy/dx = 3y^2 * 8x^3.Express in Terms of x: Next, we need to find the derivative of 4cos(x) with respect to x. The derivative of cos(x) with respect to x is -sin(x), so the derivative of 4cos(x) with respect to x is -4sin(x).Simplify Expression: Now we can combine the derivatives of y^3 and 4cos(x) to find the derivative of the entire expression y^3 + 4cos(x) with respect to x: (d/dx)(y^3 + 4cos(x)) = (d/dx)(y^3) + (d/dx)(4cos(x)) = 3y^2 * 8x^3 - 4sin(x).Finally, we need to express the derivative in terms of x only. We know that y = 2x^4 + 1, so we can substitute y in the expression 3y^2 * 8x^3 - 4sin(x) with 2x^4 + 1 to get the final derivative in terms of x: (d/dx)(y^3 + 4cos(x)) = 3(2x^4 + 1)^2 * 8x^3 - 4sin(x).Now we simplify the expression 3(2x^4 + 1)^2 * 8x^3 - 4sin(x). First, we square (2x^4 + 1) to get (2x^4 + 1)^2 = 4x^8 + 4x^4 + 1. Then we multiply this by 3 * 8x^3 to get 3 * 8x^3 * (4x^8 + 4x^4 + 1) = 24x^3 * (4x^8 + 4x^4 + 1) = 96x^11 + 96x^7 + 24x^3. Finally, we subtract 4sin(x) to get the complete expression in terms of x: 96x^11 + 96x^7 + 24x^3 - 4sin(x).

Given that $y=2 x^{4}+1$ , find $\frac{d}{d x}\left(y^{3}+4 \cos x\right)$ in terms of only $x$ . $\newline$ Answer:

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Given that y=2x4+1 y=2 x^{4}+1 y=2x4+1, find ddx(y3+4cos⁡x) \frac{d}{d x}\left(y^{3}+4 \cos x\right) dxd​(y3+4cosx) in terms of only x x x.\newlineAnswer:

Given that $y=2 x^{4}+1$ , find $\frac{d}{d x}\left(y^{3}+4 \cos x\right)$ in terms of only $x$ . $\newline$ Answer: