Let g be a function such that g(-1)=5 and g^(')(-1)=6. Let h be the function h(x)=2x^(2). Evaluate (d)/(dx)[(g(x))/(h(x))] at x=-1.

Question

Let  g be a function such that  g(-1)=5 and  g^(')(-1)=6. Let  h be the function  h(x)=2x^(2).  Evaluate  (d)/(dx)[(g(x))/(h(x))] at  x=-1.

Bytelearn · Accepted Answer

Apply Quotient Rule: First, we need to apply the quotient rule for differentiation, which is given by: (d/dx)[u(x)/v(x)] = (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2 Here, u(x) = g(x) and v(x) = h(x).Find Derivatives: We need to find the derivatives of g(x) and h(x). We are given g'(-1) = 6, so we already have the derivative of g(x) at x = -1. For h(x), we need to differentiate h(x) = 2x^2. The derivative of h(x) with respect to x is h'(x) = d/dx[2x^2] = 4x.Evaluate Derivatives: Now we evaluate h'(x) at x = -1 to get h'(-1). h'(-1) = 4*(-1) = -4.Apply Quotient Rule: Next, we need to evaluate g(x) and h(x) at x = -1. We are given g(-1) = 5, and we can calculate h(-1) = 2*(-1)^2 = 2*1 = 2.Substitute Values: Now we have all the values we need to apply the quotient rule: g'(-1) = 6, g(-1) = 5, h'(-1) = -4, and h(-1) = 2. Using the quotient rule: (d/dx)[g(x)/h(x)] at x = -1 = (h(-1) * g'(-1) - g(-1) * h'(-1)) / (h(-1))^2Perform Calculations: Substitute the known values into the quotient rule formula: (d/dx)[g(x)/h(x)] at x = -1 = (2 * 6 - 5 * -4) / (2)^2Now, perform the calculations: (d/dx)[g(x)/h(x)] at x = -1 = (12 + 20) / 4 (d/dx)[g(x)/h(x)] at x = -1 = 32 / 4 (d/dx)[g(x)/h(x)] at x = -1 = 8

- Let $g$ be a function such that $g(-1)=5$ and $g^{\prime}(-1)=6$ . $\newline$ - Let $h$ be the function $h(x)=2 x^{2}$ . $\newline$ Evaluate $\frac{d}{d x}\left[\frac{g(x)}{h(x)}\right]$ at $x=-1$ .

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- Let g g g be a function such that g(−1)=5 g(-1)=5 g(−1)=5 and g′(−1)=6 g^{\prime}(-1)=6 g′(−1)=6.\newline- Let h h h be the function h(x)=2x2 h(x)=2 x^{2} h(x)=2x2.\newlineEvaluate ddx[g(x)h(x)] \frac{d}{d x}\left[\frac{g(x)}{h(x)}\right] dxd​[h(x)g(x)​] at x=−1 x=-1 x=−1.

- Let $g$ be a function such that $g(-1)=5$ and $g^{\prime}(-1)=6$ . $\newline$ - Let $h$ be the function $h(x)=2 x^{2}$ . $\newline$ Evaluate $\frac{d}{d x}\left[\frac{g(x)}{h(x)}\right]$ at $x=-1$ .