Question

how does y=x^2(x^3-1) become 5x^4-2x using derivatives?

Answer 1

Do you know the power rule, or even the product rule for derivatives?

Answer 2

I vaguely understand it, I get confuse

Answer 3

Well, you can either use product rule or multiply them out and do power rule.
This is the power rule:
\[\frac{d}{dx}cx^n=cnx^{n-1}\]To use this, we just:
\[x^2(x^3-1)= x^5-x^2 \\ \frac{d}{dx}(x^5-x^2) =\frac{d}{dx}(x^5)-\frac{d}{dx}(x^2)=5x^4-2x \]
Then we have product rule:
\[\frac{d}{dx}(f\cdot g) =\frac{d}{dx}(f)g\cdot f\frac{d}{dx}(f)\]To use this we say
\[f=x^2, g=x^3-1\\
\frac{d}{dx}x^2(x^3-1)=\frac{d}{dx}(x^2)\cdot (x^2-1) + x^2 \frac{d}{dx}(x^3-1) = 5x^4-2x
\]

Answer 4

thank you