Given that y=4x^(2)-5, find (d)/(dx)(3x^(5)-3cos y) in terms of only x. Answer:

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Given that  y=4x^(2)-5, find  (d)/(dx)(3x^(5)-3cos y) in terms of only  x. Answer:

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Find Derivative of 3x^(5): We need to find the derivative of the function 3x^(5) - 3cos(y) with respect to x. We will use the chain rule for the term involving y, since y is a function of x. First, let's find the derivative of 3x^(5) with respect to x. The derivative of x^n with respect to x is n*x^(n-1), so the derivative of 3x^(5) is 5*3x^(5-1) = 15x^(4).Derivative of -3cos(y): Now, let's find the derivative of -3cos(y) with respect to x. 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 derivative of -3cos(y) with respect to y is 3sin(y), and we need to multiply this by the derivative of y with respect to x.Derivative of y: We are given that y = 4x^(2) - 5. Let's find the derivative of y with respect to x. The derivative of 4x^(2) with respect to x is 2*4x^(2-1) = 8x, and the derivative of a constant is 0, so the derivative of -5 is 0. Therefore, the derivative of y with respect to x is 8x.Combine Derivatives: Now we can combine the derivatives we found. The derivative of 3x^(5) - 3cos(y) with respect to x is the derivative of 3x^(5) plus the derivative of -3cos(y) times the derivative of y with respect to x. This gives us 15x^(4) - 3sin(y) * 8x.Express in Terms of x: We need to express the derivative in terms of x only. We have the term -3sin(y) * 8x, where y = 4x^(2) - 5. We substitute y into the sine function to get -3sin(4x^(2) - 5) * 8x.Final Derivative: The final derivative of the function 3x^(5) - 3cos(y) with respect to x, expressed in terms of x only, is: 15x^(4) - 24xsin(4x^(2) - 5).

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

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