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Differentiating the function $f(x) = x - rac{1}{2}(x + 6)^{ rac{1}{2}}$ with respect to $x$: To find the derivative, we apply the rules of differentiation - Leaving Cert Mathematics - Question b, c - 2015

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Differentiating-the-function-$f(x)-=-x----rac{1}{2}(x-+-6)^{-rac{1}{2}}$-with-respect-to-$x$:--To-find-the-derivative,-we-apply-the-rules-of-differentiation-Leaving Cert Mathematics-Question b, c-2015.png

Differentiating the function $f(x) = x - rac{1}{2}(x + 6)^{ rac{1}{2}}$ with respect to $x$: To find the derivative, we apply the rules of differentiation. 1. The... show full transcript

Worked Solution & Example Answer:Differentiating the function $f(x) = x - rac{1}{2}(x + 6)^{ rac{1}{2}}$ with respect to $x$: To find the derivative, we apply the rules of differentiation - Leaving Cert Mathematics - Question b, c - 2015

Step 1

Differentiate $x - rac{1}{2}(x + 6)^{ rac{1}{2}}$ with respect to $x$

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Answer

The derivative f(x)f'(x) is calculated as follows:

f'(x) = 1 - rac{1}{2(x + 6)^{ rac{1}{2}}}

Step 2

Find the co-ordinates of the turning point of the function $y = x - rac{1}{2}(x + 6)^{ rac{1}{2}}$, $x \geq -6$

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Answer

Setting the derivative to 0: 0 = 1 - rac{1}{2(x + 6)} Solving gives: x = - rac{11}{2} Now substituting back to find yy: y = - rac{11}{2} - rac{1}{2 ext{(} rac{1}{2} ext{)}} The coordinates of the turning point are ext{(}- rac{11}{2}, -5 ext{)}.

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