How do you find the slope of the tangent to the curve #y = 1/sqrtx# at the point where x = a?
You can find the derivative and evaluate it at your point:
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To find the slope of the tangent to the curve (y = \frac{1}{\sqrt{x}}) at the point where (x = a), you use differentiation to find the derivative of (y) with respect to (x), which gives the slope of the tangent line at any point (x) on the curve.
Given (y = \frac{1}{\sqrt{x}}), we can rewrite (y) as (x^{-\frac{1}{2}}).
The derivative of (y) with respect to (x), denoted as (\frac{dy}{dx}) or (y'), is found using the power rule for differentiation. The power rule states that if (y = x^n), then (y' = nx^{n-1}).
Applying this rule to (y = x^{-\frac{1}{2}}), we get:
[y' = -\frac{1}{2}x^{-\frac{1}{2} - 1} = -\frac{1}{2}x^{-\frac{3}{2}}]
This derivative gives us the slope of the tangent at any point (x) on the curve. To find the slope at (x = a), substitute (x) with (a):
[y' = -\frac{1}{2}a^{-\frac{3}{2}}]
Therefore, the slope of the tangent to the curve (y = \frac{1}{\sqrt{x}}) at the point where (x = a) is (-\frac{1}{2}a^{-\frac{3}{2}}).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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