mark
Use the definition of the derivative to find the derivative of
f(x) =x^5.
An archived instance of a Calc I forum
Derivative of a power function
mearing
\lim_{h \to 0}\frac{(x+h)^5-(x)^5}{h}
\lim_{h \to 0}\frac{(x^5+5x^4h+10x^3h^2+10x^2h^3+5h^4x+h^5)-x^5}{h}
\lim_{h \to 0}\frac{h(5x^4+10x^3h+10x^2h^2+5h^3x+h^4)}{h}
\lim_{h \to 0} (5x^4+10x^3h+10x^2h^2+5h^3x+h^4)
f'(x)= 5x^4
jbarry1
f^{\prime }(x)=\frac{\left(\left(x+h\right)^5-x^5\right)}{h}
f^{\prime }(x)=\frac{\left(x^5+5x^4h+10x^3h^2+10x^2h^3+5xh^4+h^5-x^5\right)}{h}
f^{\prime }(x)=\left(\frac{h(5x^4+10x^3h+10x^2h^2+5xh^3+h^4)}{h}\right)
f^{\prime }(x)=\lim _{h\to 0}\left(5x^4+10x^3\left(0\right)+10x^2\left(0\right)^2+5x\left(0\right)^3+\left(0\right)^4\right)=5x^4