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Algebra of Derivatives

If m is the s...

Question

If $m$ is the slope of a tangent to the curve $e^{y} = 1 + x^{2},$ then

| m | > 1

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m > 1

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m > - 1

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| m | \leq 1

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Solution

The correct option is D $| m | \leq 1$
$\frac{d y}{d x} = m = \frac{2 x}{1 + x^{2}}$
$\Rightarrow | m | = \frac{2 | x |}{1 + x^{2}}$
$(1 - x)^{2} \geq 0$
$\Rightarrow 1 + x^{2} \geq 2 x$
$(1 + x)^{2} \geq 0$
$\Rightarrow 1 + x^{2} \geq - 2 x$
$∴ 1 + x^{2} \geq 2 | x |$
$\Rightarrow | m | \leq 1$

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