Given the function y=(-6-10x^(-1)-x)(-7-7x^(3)), find (dy)/(dx) in any form.

Question

Given the function  y=(-6-10x^(-1)-x)(-7-7x^(3)), find  (dy)/(dx) in any form.

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Identify functions: We are given the function y=(-6-10x^(-1)-x)(-7-7x^(3)). To find the derivative of this function with respect to x, we will use the product rule, which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.Derivative of u: First, let's identify the two functions that are being multiplied. We have u = -6-10x^(-1)-x and v = -7-7x^(3). We will need to find the derivatives of both u and v with respect to x.Derivative of v: The derivative of u with respect to x is given by: (du)/(dx) = (d)/(dx)(-6) + (d)/(dx)(-10x^(-1)) + (d)/(dx)(-x) The derivative of a constant is 0, so (d)/(dx)(-6) = 0. The derivative of -10x^(-1) is 10x^(-2) because (d)/(dx)(x^n) = nx^(n-1). The derivative of -x is -1 because (d)/(dx)(x) = 1. So, (du)/(dx) = 0 + 10x^(-2) - 1.Apply product rule: Now, let's find the derivative of v with respect to x: (dv)/(dx) = (d)/(dx)(-7) + (d)/(dx)(-7x^(3)) Again, the derivative of a constant is 0, so (d)/(dx)(-7) = 0. The derivative of -7x^(3) is -21x^(2) because (d)/(dx)(x^n) = nx^(n-1). So, (dv)/(dx) = 0 - 21x^(2).Distribute terms: Now we can apply the product rule: (dy)/(dx) = (du)/(dx) * v + u * (dv)/(dx) Substituting the derivatives we found: (dy)/(dx) = (10x^(-2) - 1) * (-7-7x^(3)) + (-6-10x^(-1)-x) * (-21x^(2))Combine like terms: Let's distribute the terms: (dy)/(dx) = -70x^(-2) + 7 + 70x + 7x^(4) - 126x - 210 - 21x^(3)Now we combine like terms: (dy)/(dx) = 7x^(4) - 21x^(3) - 56x - 203 + 7x^(-2)

Given the function $y=\left(-6-10 x^{-1}-x\right)\left(-7-7 x^{3}\right)$ , find $\frac{d y}{d x}$ in any form.

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Given the function y=(−6−10x−1−x)(−7−7x3) y=\left(-6-10 x^{-1}-x\right)\left(-7-7 x^{3}\right) y=(−6−10x−1−x)(−7−7x3), find dydx \frac{d y}{d x} dxdy​ in any form.

Given the function $y=\left(-6-10 x^{-1}-x\right)\left(-7-7 x^{3}\right)$ , find $\frac{d y}{d x}$ in any form.