Question

The equation of a curve is y= (1+x)/(1+2x) for x> -1/2. Show that the gradient of the curve is always negative.​

Answer 1

Step-by-step explanation:

Hope u will understand

Answer 2

Given,

Equation of curve,

$\[y = \frac{{1 + x}}{{1 + 2x}},x > - \frac{1}{2}\]$

Solution,

Calculate the gradient of equation of curve.

$\[ \Rightarrow \frac{{dy}}{{dx}} = \frac{{1 + 2x - \left( {1 + x} \right) \times 2}}{{{{\left( {1 + 2x} \right)}^2}}}\]$

$\[ \Rightarrow \frac{{dy}}{{dx}} = \frac{{1 + 2x - 2 - 2x}}{{{{\left( {1 + 2x} \right)}^2}}}\]$

$\[ \Rightarrow \frac{{dy}}{{dx}} = \frac{{ - 1}}{{{{\left( {1 + 2x} \right)}^2}}}\]$

$\[ \Rightarrow \frac{{dy}}{{dx}} = - ve\]$

The denominator is always positive because it is a square term. Denominator is independent of sign of x. Numerator is 1 which is constant. So gradient is always negative irrespective of sign of x value.