Question

Find the derivative of the function using the definition of derivative.

G(t) = (1-2t) / 6+t
State the domain of the function. (Enter your answer using interval notation.)
State the domain of its derivative. (Enter your answer using interval notation.)

Answer 1

To find the derivative of the function G(t) = (1-2t)/(6+t) using the definition of derivative, apply the quotient rule. The derivative is G'(t) = 10/(6+t)^2. The domain of G(t) and G'(t) is (-∞, -6) ∪ (-6, ∞).

Step-by-step explanation:

To find the derivative of the function G(t) = (1-2t)/(6+t) using the definition of derivative, we can apply the quotient rule. The quotient rule states that if we have a function f(t) = g(t)/h(t), the derivative of f(t) is given by f'(t) = (g'(t)h(t) - g(t)h'(t))/(h(t))^2. So in this case, the derivative of G(t) is:

G'(t) = [(2(6+t) - (1-2t)(1))/(6+t)^2]

Simplifying this expression gives G'(t) = 10/(6+t)^2.

The domain of the function G(t) is the set of all real numbers except -6, since dividing by zero is undefined. So the domain is (-∞, -6) ∪ (-6, ∞).

The domain of the derivative G'(t) is also the set of all real numbers except -6, so the domain is the same as that of the original function, (-∞, -6) ∪ (-6, ∞).