Given the function y=(-8x-10)(-6x^(3)-8-7x^(-1)), find (dy)/(dx) in any form. Answer: (dy)/(dx)=

Question

Given the function  y=(-8x-10)(-6x^(3)-8-7x^(-1)), find  (dy)/(dx) in any form. Answer:  (dy)/(dx)=

Bytelearn · Accepted Answer

Apply Product Rule: To find the derivative of the given function, 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.Find Derivatives: Let's denote the first function as f(x) = -8x - 10 and the second function as g(x) = -6x^3 - 8 - 7x^(-1). We will first find the derivatives f'(x) and g'(x).Calculate f'(x): The derivative of f(x) = -8x - 10 with respect to x is f'(x) = -8, since the derivative of a constant is 0 and the derivative of -8x with respect to x is -8.Calculate g'(x): The derivative of g(x) = -6x^3 - 8 - 7x^(-1) with respect to x is g'(x) = -18x^2 + 7x^(-2). This is because the derivative of -6x^3 is -18x^2, the derivative of -8 is 0, and the derivative of -7x^(-1) is 7x^(-2).Use Product Rule: Now we apply the product rule: (dy)/(dx) = f'(x)g(x) + f(x)g'(x).Substitute Expressions: Substitute the expressions for f'(x), g(x), f(x), and g'(x) into the product rule formula: (dy)/(dx) = (-8)(-6x^3 - 8 - 7x^(-1)) + (-8x - 10)(-18x^2 + 7x^(-2)).Simplify Expression: Now we simplify the expression: (dy)/(dx) = 48x^3 + 64 + 56x^(-1) - 144x^3 - 80x^2 - 56x - 70x^(-2).Combine Like Terms: Combine like terms to get the final derivative: (dy)/(dx) = (48x^3 - 144x^3) + (-80x^2) + (56x^(-1) - 56x) + (64 - 70x^(-2)).Final Derivative: Simplify the expression further: (dy)/(dx) = -96x^3 - 80x^2 - 56x + 56x^(-1) + 64 - 70x^(-2).Simplify Further: The final simplified form of the derivative is: (dy)/(dx) = -96x^3 - 80x^2 - 56x + 56/x + 64 - 70/x^2.

Given the function $y=(-8 x-10)\left(-6 x^{3}-8-7 x^{-1}\right)$ , find $\frac{d y}{d x}$ in any form. $\newline$ Answer: $\frac{d y}{d x}=$

Full solution

More problems from Find derivatives of using multiple formulae

Related topics

Given the function y=(−8x−10)(−6x3−8−7x−1) y=(-8 x-10)\left(-6 x^{3}-8-7 x^{-1}\right) y=(−8x−10)(−6x3−8−7x−1), find dydx \frac{d y}{d x} dxdy​ in any form.\newlineAnswer: dydx= \frac{d y}{d x}= dxdy​=

Given the function $y=(-8 x-10)\left(-6 x^{3}-8-7 x^{-1}\right)$ , find $\frac{d y}{d x}$ in any form. $\newline$ Answer: $\frac{d y}{d x}=$