the derivative as a function\nquestion\ncompleted: 0 of 6 | my score: 0/6 pts (0%\nuse the formula…

the derivative as a function\nquestion\ncompleted: 0 of 6 | my score: 0/6 pts (0%\nuse the formula f(x)=\\lim_{z\\to x}\\frac{f(z)-f(x)}{z - x} to find the derivative of the following function.\nf(x)=\\frac{3}{x + 10}\nf(x)=\\square

the derivative as a function\nquestion\ncompleted: 0 of 6 | my score: 0/6 pts (0%\nuse the formula f(x)=\\lim_{z\\to x}\\frac{f(z)-f(x)}{z - x} to find the derivative of the following function.\nf(x)=\\frac{3}{x + 10}\nf(x)=\\square

Answer

Explanation:

Step1: Find f(z)

Given (f(x)=\frac{3}{x + 10}), then (f(z)=\frac{3}{z + 10}).

Step2: Substitute f(z) and f(x) into the derivative formula

[ \begin{align*} f^{\prime}(x)&=\lim_{z\rightarrow x}\frac{f(z)-f(x)}{z - x}\ &=\lim_{z\rightarrow x}\frac{\frac{3}{z + 10}-\frac{3}{x + 10}}{z - x}\ &=\lim_{z\rightarrow x}\frac{\frac{3(x + 10)-3(z + 10)}{(z + 10)(x + 10)}}{z - x}\ &=\lim_{z\rightarrow x}\frac{3x+30 - 3z-30}{(z - x)(z + 10)(x + 10)}\ &=\lim_{z\rightarrow x}\frac{3x-3z}{(z - x)(z + 10)(x + 10)}\ &=\lim_{z\rightarrow x}\frac{-3(z - x)}{(z - x)(z + 10)(x + 10)} \end{align*} ]

Step3: Simplify the limit

Cancel out the ((z - x)) terms: [ \begin{align*} f^{\prime}(x)&=\lim_{z\rightarrow x}\frac{-3}{(z + 10)(x + 10)}\ &=\frac{-3}{(x + 10)(x + 10)}\ &=-\frac{3}{(x + 10)^2} \end{align*} ]

Answer:

(-\frac{3}{(x + 10)^2})