Question

quiz instructions
access code: start
timed: 25 minutes
number of attempts: 2
content: review of power rule, quotient rule, derivative of trig functions, and related rates
question 2
find f in terms of g:
f(x)=x^{3}g(x)
f(x)=3x^{2}f(x)+x^{3}g(x)
f(x)=3x^{2}g(x)+x^{3}g(x)
f(x)=3x^{2}g(x)
f(x)=3x^{2}g(x)+3x^{3}g(x)
f(x)=3x^{2}+g(x)

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u(x)v(x)$, then $y'=u'(x)v(x)+u(x)v'(x)$. Here, $u(x)=x^{3}$ and $v(x)=g(x)$.

Step2: Find derivative of $u(x)$

The derivative of $u(x)=x^{3}$ using the power - rule ($(x^{n})'=nx^{n - 1}$) is $u'(x)=3x^{2}$.

Step3: Calculate $f'(x)$

By the product - rule, $f'(x)=(x^{3})'g(x)+x^{3}g'(x)=3x^{2}g(x)+x^{3}g'(x)$.

Answer:

$f'(x)=3x^{2}g(x)+x^{3}g'(x)$ (the second option)