How do you use the definition of a derivative to find the derivative of $f(x) = -6 – x^2$ at x=7?

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Answer 1 · 2018-01-27T06:39:16

At $x=7$ , $(df)/(dx)=-14$

For a function $f(x)$ , its derivative $(df)/(dx)$ is defined as

$(df)/(dx)=lim_(h->0)(f(x+h)-f(x))/h$

As $f(x)=-6-x^2$ , $f(x+h)=-6-(x+h)^2$

and $(df)/(dx)=lim_(h->0)(f(x+h)-f(x))/h$

= $lim_(h->0)(-6-(x+h)^2-(-6-x^2))/h$

= $lim_(h->0)(-6-x^2-2hx-h^2+6+x^2)/h$

= $lim_(h->0)(-2hx-h^2)/h$

= $lim_(h->0)(-2x-h)$

= $-2x$

and at $x=7$ , we have $(df)/(dx)=-2xx7=-14$

How do you use the definition of a derivative to find the derivative of f(x) = -6 – x^2 f(x) = -6 – x^2 at x=7?