Question

Identify each expression that represents the slope of a tangent to the curve y = 1/x+1 at any point (x,y)

Answer 1

The expressions that represent the slope of a tangent to the curve $y = \frac{1}{x+1}$ are $m = \lim_{h \to 0}\frac{\frac{1}{x + h + 1} - \frac{1}{x + 1} }{h}$ and $m = -\frac{1}{(x + 1)^{2}}$, respectively.

How to find an expression for the slope at any point of a function

The slope at any point of the function can be found by definition of derivative following algebraic handling:

$m = \lim_{h \to 0}\frac{\frac{1}{x + h + 1} - \frac{1}{x + 1} }{h}$

$m = \lim_{h \to 0} \frac{\frac{x+1 - x - h - 1}{(x + h + 1)\cdot (x + 1)} }{h}$

$m = -\lim_{h\to 0} \frac{1}{(x + h + 1)\cdot (x + 1)}$

$m = -\frac{1}{(x + 1)^{2}}$

Thus, the expressions that represent the slope of a tangent to the curve $y = \frac{1}{x+1}$ are $m = \lim_{h \to 0}\frac{\frac{1}{x + h + 1} - \frac{1}{x + 1} }{h}$ and $m = -\frac{1}{(x + 1)^{2}}$, respectively.

To learn more on differentiation: brainly.com/question/18962394

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