Question

derive y= -x^2 (1+x^2-x^3 ) using the definition of derivative? please help me..

Answer 1

@mukushla

Answer 2

Two ways: one is to multiply it out and then use the power rule, the other is to use the product rule. Do you know the product rule?

Answer 3

Let u = -x^2
v = (1+x^2-x^3 )
what are du and dv?

Answer 4

Just for reference, the product rule is this:
d/dx (uv) = u dv/dx + v du/dx
or more simply
(uv)' = u v' + v u'

Answer 5

can you find u' if u = -x^2 ?

Answer 6

\(f(x)=-x^2-x^4+x^5 \)

by definition

\[f'(x)=\lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}\\=\lim_{\Delta x \rightarrow 0} \frac{-(x+\Delta x)^2-(x+\Delta x)^4+(x+\Delta x)^5+x^2+x^4-x^5}{\Delta x}\]

Answer 7

simplifying and neglecting the terms which consists exponents of \(\Delta x\) with powers greater than or equal to 2 u will have...

\[f'(x)=\lim_{\Delta x \rightarrow 0} \frac{-2x\Delta x-4x^3 \Delta x+5x^4\Delta x}{\Delta x}=-2x-4x^3+5x^4\]