Final answer:
To find g'(-1/5), differentiate the given equation with respect to x step by step. Substitute g(x) = 0 and x = -1/5, then solve for g'(-1/5). The answer is d) g'(-1/5) = -63/40.
Step-by-step explanation:
To find g'(-1/5), we need to differentiate the given equation with respect to x. Let's do this step by step:
First, differentiate 8g(x) with respect to x. Since g(x) is a function of x, its derivative is g'(x). So, 8g'(x).
Next, differentiate -3x sin(g(x)) with respect to x using the product rule. The derivative of -3x with respect to x is -3 and the derivative of sin(g(x)) with respect to x is cos(g(x))g'(x). So, -3cos(g(x))g'(x).
Finally, differentiate 20x^2 - 11x - 3 with respect to x. The derivative of 20x^2 is 40x, the derivative of -11x is -11, and the derivative of -3 is 0. So, we have 40x - 11.
Now, let's substitute g(x) = 0 and x = -1/5 into the equation and solve for g'(-1/5):
8g'(-1/5) - 3cos(g(-1/5))g'(-1/5) = 40(-1/5) - 11.
Since g(-1/5) = 0, the equation simplifies to 8g'(-1/5) = -8/5 - 11. Combining like terms, we get 8g'(-1/5) = -8/5 - 55/5. Simplifying further, we have 8g'(-1/5) = -63/5.
To find g'(-1/5), divide both sides of the equation by 8: g'(-1/5) = (-63/5) / 8.
Simplifying the right side of the equation, we have g'(-1/5) = -63/40.
Therefore, the answer is d) g'(-1/5) = -63/40.