Question

The function f (x) = x2 + x , is differentiable at x = −1

Answer 1

Answer on Question #84076 – Math – Real Analysis

Question

The function $f(x) = x^2 + x$ is differentiable at $x = -1$.

Solution

A function is differentiable at a point if it has a derivative there. In other words, the function $f$ is differentiable at $x$ if

$$\lim_{h \to 0} \frac{(f(x + h) - f(x))}{h}$$

exists.

Find a limit

$$\lim_{h \to 0} \frac{(x + h)^2 + (x + h) - ((x)^2 + (x))}{h} =$$ $$= \lim_{h \to 0} \frac{x^2 + 2xh + h^2 + x + h - x^2 - x}{h} = \lim_{h \to 0} \frac{2xh + h^2 + h}{h} = 2x + 1.$$ $$f'(x) = 2x + 1.$$

The function $f(x) = x^2 + x$ is differentiable at $x = -1$:

$$f'(-1) = 2^*(-1) + 1 = -2 + 1 = -1;$$ $$f'(-1) = -1.$$

**Answer:** Yes, the function $f(x) = x^2 + x$ is differentiable at $x = -1$.

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