Question

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^6g(x)
○ f(x)=6x^5 + g(x)
○ f(x)=6x^5g(x)
○ f(x)=6x^5g(x)+6x^6g(x)
○ f(x)=6x^5f(x)+x^6g(x)
○ f(x)=6x^5g(x)+x^6g(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, $u(x)=x^{6}$ and $v(x)=g(x)$.

Step2: Differentiate $u(x)$

The derivative of $u(x)=x^{6}$ using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ is $u'(x)=6x^{5}$. And $v'(x)=g'(x)$.

Step3: Calculate $f'(x)$

By the product - rule, $f'(x)=u'(x)v(x)+u(x)v'(x)=6x^{5}g(x)+x^{6}g'(x)$.

Answer:

$f'(x)=6x^{5}g(x)+x^{6}g'(x)$