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Bueno, la definición, tenemos,
![\displaystyle f'(x)=\lim_{h\rightarrow0}{\frac{f(x+h)-f(x)}{h}}\\\\f'(x)=\lim_{h\rightarrow0}{\frac{9(x+h)^{2}-9x^{2}}{h}}=\lim_{h\rightarrow0}{\frac{9(x^{2}+2xh+h^{2})-9x^{2}}{h}}=\\\\\lim_{h\rightarrow0}{\frac{18xh+h^{2}}{h}}=\lim_{h\rightarrow0}{\frac{h(18x+h)}{h}}=\lim_{h\rightarrow0}{18x+h}=18x+(0)=18x\\\\f'(x)=18x \displaystyle f'(x)=\lim_{h\rightarrow0}{\frac{f(x+h)-f(x)}{h}}\\\\f'(x)=\lim_{h\rightarrow0}{\frac{9(x+h)^{2}-9x^{2}}{h}}=\lim_{h\rightarrow0}{\frac{9(x^{2}+2xh+h^{2})-9x^{2}}{h}}=\\\\\lim_{h\rightarrow0}{\frac{18xh+h^{2}}{h}}=\lim_{h\rightarrow0}{\frac{h(18x+h)}{h}}=\lim_{h\rightarrow0}{18x+h}=18x+(0)=18x\\\\f'(x)=18x](https://tex.z-dn.net/?f=%5Cdisplaystyle+f%27%28x%29%3D%5Clim_%7Bh%5Crightarrow0%7D%7B%5Cfrac%7Bf%28x%2Bh%29-f%28x%29%7D%7Bh%7D%7D%5C%5C%5C%5Cf%27%28x%29%3D%5Clim_%7Bh%5Crightarrow0%7D%7B%5Cfrac%7B9%28x%2Bh%29%5E%7B2%7D-9x%5E%7B2%7D%7D%7Bh%7D%7D%3D%5Clim_%7Bh%5Crightarrow0%7D%7B%5Cfrac%7B9%28x%5E%7B2%7D%2B2xh%2Bh%5E%7B2%7D%29-9x%5E%7B2%7D%7D%7Bh%7D%7D%3D%5C%5C%5C%5C%5Clim_%7Bh%5Crightarrow0%7D%7B%5Cfrac%7B18xh%2Bh%5E%7B2%7D%7D%7Bh%7D%7D%3D%5Clim_%7Bh%5Crightarrow0%7D%7B%5Cfrac%7Bh%2818x%2Bh%29%7D%7Bh%7D%7D%3D%5Clim_%7Bh%5Crightarrow0%7D%7B18x%2Bh%7D%3D18x%2B%280%29%3D18x%5C%5C%5C%5Cf%27%28x%29%3D18x)
y eso sería todo
y eso sería todo
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