Can some one walk me through how to find the derivative of $y=6xe^{6x}-e^{6x}$?
I know the answer should be $36e^{6x}x$.
I know I am supposed to use the product and chain rule, I just don't understand where and how to tell they are applied. How do I know what part requires what?
2 Answers
$\begingroup$Hint
Rewrite as
$$y=(6x-1)e^{6x}$$
call $f(x)=6x-1$ and $g(x)=e^{6x}$ and use
$$(f\cdot g)'=f'\cdot g+f\cdot g'$$
In order to calculate $g'(x)$ you will need chain rule.
Then call $p(x)=6x$ and $q(x)=e^x$ and then $g(x)=q(p(x))$. Now use chain rule
$$g'(x)=p'(x)\cdot q'(p(x))$$
$\endgroup$ $\begingroup$Linearity of derivatives: $$\frac{d}{dx}(cf(x)+kg(x))=cf'(x)+kg'(x)\qquad c,k\in\Bbb R$$ Product rule: $$\frac{d}{dx}(f(x)g(x))=f'(x)g(x)+f(x)g'(x)$$ Chain rule: $$\frac{d}{dx}(f(g(x))=f'(g(x))g'(x)$$ Recall, $$\frac{d}{du}e^u=e^u$$
$\endgroup$
