Question

Find,from the definition,the derivative function f(x)=kx^3 where k is a constant

Answer 1

y'=df(x)/dx=3kx^2

Answer 2

\[\lim_{h\rightarrow 0}\frac{(k(x+h))^{3}-kx^{3}}{h} = \frac{kx^{3}+3kx^{2}h+3kxh^{2}+kh^{3}-kx^{3}}{h} \]
\[=\frac{kh(3x^{2}+3xh+h^{2})}{h} = 3kx^{2}+ 3xh+h^{2}\]
when you make h = 0 you just get 3kx^2

Answer 3

\[\lim_{h \rightarrow 0}\frac{k(x+h)^3-kx^3}{h}=k \lim_{h \rightarrow 0}\frac{(x^3+3x^2h+3xh^2+h^3)-x^3}{h}\]
\[k \lim_{h \rightarrow 0}\frac{3x^2h+3xh^2+h^3}{h}=k \lim_{h \rightarrow 0}\frac{3x^2+3xh+h^2}{1}=k*3x^2\]

Answer 4

thanx