find a simplified difference quotient for the function. and evaluate the derivative at a given x - value…

Explanation:

Step1: Recall difference - quotient formula

The difference - quotient formula is $\frac{f(x + h)-f(x)}{h}$, where $f(x)=-3x^{2}$. First, find $f(x + h)$: [ \begin{align*} f(x + h)&=-3(x + h)^{2}\ &=-3(x^{2}+2xh+h^{2})\ &=-3x^{2}-6xh - 3h^{2} \end{align*} ]

Step2: Calculate the difference - quotient

Substitute $f(x + h)$ and $f(x)$ into the difference - quotient formula: [ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{(-3x^{2}-6xh - 3h^{2})-(-3x^{2})}{h}\ &=\frac{-3x^{2}-6xh - 3h^{2}+3x^{2}}{h}\ &=\frac{-6xh - 3h^{2}}{h}\ &=-6x-3h \end{align*} ]

Step3: Find the derivative as $h\to0$

The derivative $f^\prime(x)=\lim_{h\to0}\frac{f(x + h)-f(x)}{h}$. Taking the limit of the difference - quotient as $h\to0$, we get $f^\prime(x)=-6x$.

Step4: Evaluate the derivative at $x = 1$

Substitute $x = 1$ into $f^\prime(x)$: $f^\prime(1)=-6\times1=-6$

Answer:

The simplified difference - quotient is $-6x - 3h$, and the value of the derivative at $x = 1$ is $-6$.