Find the derivative of the function using the definition of derivative. g(t) = 9 t g'(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.)
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Answer:domain [ g(t) ] = (-∞,∞)g'(t)=9domain [ g'(t) ] =(-∞,∞)Step-by-step explanation:We start by finding the domain of the function g(t)The domain of a function is the set of all inputs over which the function has defined outputs.In g(t) = 9t ; g(t) is define for all real numbersdomain [ g(t) ] = (-∞,∞)For the derivative of the function we use the definition of derivative :Given f(x)→[tex]f'(x) = \lim_{h \to \00} \frac{f(x+h)-f(x)}{h}[/tex]In our exercise :[tex]g'(t)= \lim_{h \to \00} \frac{g(t+h)-g(t)}{h}[/tex][tex]\lim_{h \to \00} \frac{9(t+h)-9t}{h} =\\ \lim_{h \to \00} \frac{9t+9h-9t}{h} =\\\lim_{h \to \00} \frac{9h}{h}\\\lim_{h \to \00} 9=9[/tex][tex]g'(t)=9[/tex]domain [ g'(t) ] =(-∞,∞)