by determining $f(x)=lim_{h \to 0}\frac{f(x + h)-f(x)}{h}$, find $f(8)$ for the given function\n$f(x)=6x^{2}$…

by determining $f(x)=lim_{h \to 0}\frac{f(x + h)-f(x)}{h}$, find $f(8)$ for the given function\n$f(x)=6x^{2}$\n$f(8)=square$ (simplify your answer.)

by determining $f(x)=lim_{h \to 0}\frac{f(x + h)-f(x)}{h}$, find $f(8)$ for the given function\n$f(x)=6x^{2}$\n$f(8)=square$ (simplify your answer.)

Answer

Explanation:

Step1: Find f(x + h)

Given (f(x)=6x^{2}), then (f(x + h)=6(x + h)^{2}=6(x^{2}+2xh+h^{2})=6x^{2}+12xh + 6h^{2})

Step2: Substitute into the derivative formula

[ \begin{align*} f^{\prime}(x)&=\lim_{h\rightarrow0}\frac{f(x + h)-f(x)}{h}\ &=\lim_{h\rightarrow0}\frac{(6x^{2}+12xh + 6h^{2})-6x^{2}}{h}\ &=\lim_{h\rightarrow0}\frac{6x^{2}+12xh + 6h^{2}-6x^{2}}{h}\ &=\lim_{h\rightarrow0}\frac{12xh+6h^{2}}{h}\ &=\lim_{h\rightarrow0}(12x + 6h) \end{align*} ]

Step3: Evaluate the limit

As (h\rightarrow0), (f^{\prime}(x)=12x)

Step4: Find f'(8)

Substitute (x = 8) into (f^{\prime}(x)), so (f^{\prime}(8)=12\times8 = 96)

Answer:

96