Leaving Cert Mathematics Differentiation: Differentiating the function $f(

Question

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 derivative of $x$ with respect to $x$ is 1.
2. For the term $-rac{1}{2}(x + 6)^{rac{1}{2}}$, we use the chain rule:
   - The outer function is $u^{rac{1}{2}}$, where $u = x + 6$.
   - The derivative of $u^{rac{1}{2}}$ is $rac{1}{2}u^{-rac{1}{2}} * rac{du}{dx}$, where $rac{du}{dx} = 1$.

Therefore, we find:

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

To find the co-ordinates of the turning point of the function $y = x - rac{1}{2}(x + 6)^{rac{1}{2}}$, for $x ightarrow -6$:

Setting the derivative $f'(x)$ to 0, we solve:
$$0 = 1 - rac{1}{2	ext{(}x + 6	ext{)}} \
=> 2	ext{(}x + 6	ext{)} = 1 \
=> 2x + 12 = 1 \
=> 2x = -11 \
=> x = -rac{11}{2}$$

Now substituting $x$ back into the function to find $y$:
$$y = -rac{11}{2} - rac{1}{2} 	ext{(}-rac{11}{2} + 6	ext{)}^{rac{1}{2}} \
 = -rac{11}{2} - rac{1}{2} 	ext{(}rac{1}{2}	ext{)}^{rac{1}{2}} \
 = -rac{11}{2} - rac{1}{2	ext{(}rac{1}{2}	ext{)}} = -rac{11}{2} - rac{1}{2 	imes rac{1}{2}} = -rac{11}{2} + 1 = -rac{11}{2} + 1 = -rac{10}{2} = -5.
$$
Therefore, the coordinates are $	ext{(}-rac{11}{2}, -5	ext{)}$.

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

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

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

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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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

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

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

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