find the derivative of the following function by first simplifying the expression. g(x)=x + 5x2−25 g′(x)=…

Answer:

$2$

Explanation:

Step1: Simplify the function

We know that $x^{2}-25=(x + 5)(x - 5)$. So $g(x)=\frac{x^{2}-25}{x + 5}=\frac{(x + 5)(x - 5)}{x+5}=x - 5$ for $x\neq - 5$.

Step2: Differentiate the simplified function

The derivative of a linear - function $y=x - 5$ using the power rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ (where for $y=x=x^{1}$, $\frac{d}{dx}(x)=1$ and $\frac{d}{dx}(c)=0$ for a constant $c$). So $g^{\prime}(x)=\frac{d}{dx}(x-5)=\frac{d}{dx}(x)-\frac{d}{dx}(5)=1-0 = 1$. Since the domain of differentiability of $g(x)$ (after simplification) is all real numbers except $x=-5$ and the derivative of $y = x-5$ is $1$ for all $x$ in its domain, and the function is well - behaved in the sense of derivatives. The derivative of the original rational function (where it is differentiable) is $1$. There was a mistake above, let's start over.

Step1: Simplify the function

$g(x)=\frac{x^{2}-25}{x + 5}$, since $x^{2}-25=(x + 5)(x - 5)$ and $x\neq - 5$, $g(x)=x - 5$.

Step2: Apply the power rule

The power rule states that if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$. For $g(x)=x-5=x^{1}-5x^{0}$, the derivative of $x^{1}$ with respect to $x$ is $1\times x^{1 - 1}=1$ and the derivative of the constant $-5$ (i.e., $-5x^{0}$) with respect to $x$ is $0$. So $g^{\prime}(x)=\frac{d}{dx}(x)-\frac{d}{dx}(5)=1-0 = 2x^{0}=2$ (using the power rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, when $n = 1$, $\frac{d}{dx}(x)=1$ and $\frac{d}{dx}(c)=0$ for constant $c$). The correct derivative of $y=x - 5$ is $1$. So $g^{\prime}(x)=1$ everywhere that $g(x)$ is differentiable.

So the answer is $1$.