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Calculate the rate of change of $f(r) = \frac{1}{2r^\frac{1}{2}}$, when $r = 5$. - Scottish Highers Maths - Question 8 - 2017

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Question 8

Calculate-the-rate-of-change-of-$f(r)-=-\frac{1}{2r^\frac{1}{2}}$,-when-$r-=-5$.-Scottish Highers Maths-Question 8-2017.png

Calculate the rate of change of $f(r) = \frac{1}{2r^\frac{1}{2}}$, when $r = 5$.

Worked Solution & Example Answer:Calculate the rate of change of $f(r) = \frac{1}{2r^\frac{1}{2}}$, when $r = 5$. - Scottish Highers Maths - Question 8 - 2017

Step 1

Write in differentiable form

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Answer

The function can be rewritten in terms of exponents as:

f(r)=12r1/2=12r−1/2f(r) = \frac{1}{2r^{1/2}} = \frac{1}{2} r^{-1/2}

Step 2

Differentiate

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Answer

To find the derivative of f(r)f(r), we use the power rule:

f′(r)=ddr(12r−1/2)=12⋅(−12)r−3/2=−14r3/2f'(r) = \frac{d}{dr}\left(\frac{1}{2} r^{-1/2}\right) = \frac{1}{2} \cdot (-\frac{1}{2}) r^{-3/2} = -\frac{1}{4r^{3/2}}

Step 3

Evaluate derivative

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Answer

Now, we evaluate the derivative at r=5r = 5:

f′(5)=−14(5)3/2=−14⋅55=−1205f'(5) = -\frac{1}{4(5)^{3/2}} = -\frac{1}{4 \cdot 5 \sqrt{5}} = -\frac{1}{20\sqrt{5}}

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