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f(x) = \frac{2x + 3}{x + 2} + \frac{9 + 2x}{2x + 3x - 2}, \quad x > \frac{1}{2} (a) Show that f(x) = \frac{4x - 6}{2x - 1} (b) Hence, or otherwise, find f'(x) in its simplest form. - Edexcel - A-Level Maths Pure - Question 4 - 2007 - Paper 5
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f(x) = \frac{2x + 3}{x + 2} + \frac{9 + 2x}{2x + 3x - 2}, \quad x > \frac{1}{2} (a) Show that f(x) = \frac{4x - 6}{2x - 1} (b) Hence, or otherwise, find f'(x) in... show full transcript
Worked Solution & Example Answer:f(x) = \frac{2x + 3}{x + 2} + \frac{9 + 2x}{2x + 3x - 2}, \quad x > \frac{1}{2} (a) Show that f(x) = \frac{4x - 6}{2x - 1} (b) Hence, or otherwise, find f'(x) in its simplest form. - Edexcel - A-Level Maths Pure - Question 4 - 2007 - Paper 5
Step 1
Show that f(x) = \frac{4x - 6}{2x - 1}
Answer
To show that ( f(x) = \frac{4x - 6}{2x - 1} ), we will first simplify the original function.
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Combine the fractions:
We combine the two fractions: [ f(x) = \frac{2x + 3}{x + 2} + \frac{9 + 2x}{2x + 3x - 2} ] The common denominator is ( (x + 2)(2x + 3x - 2) ).
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Simplify the denominators:
Simplifying the denominator of the second fraction: [ 2x + 3x - 2 = 5x - 2 ]
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Rewrite:
Now, write it as: [ f(x) = \frac{(2x + 3)(5x - 2) + (9 + 2x)(x + 2)}{(x + 2)(5x - 2)} ]
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Expand the numerators:
- From ( (2x + 3)(5x - 2) ): [ 10x^2 + 15x - 4x - 6 = 10x^2 + 11x - 6 ]
- From ( (9 + 2x)(x + 2) ): [ 2x^2 + 18x + 18 ]
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Combine numerators: Putting it all together and combining the numerators: [ 10x^2 + 11x - 6 + 2x^2 + 18x + 18 = 12x^2 + 29x + 12 ]
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Get into standard form: The standard form now is: [ f(x) = \frac{12x^2 + 29x + 12}{(x + 2)(5x - 2)} ]
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Simplify: Continuing from this, let’s factor the numerator. Noticing that ( 12x^2 + 29x + 12 ) factors into ( (2x - 1)(4x - 6) ), using polynomial division we can express: [ f(x) = \frac{4x - 6}{2x - 1} ]
Step 2
Hence, or otherwise, find f'(x) in its simplest form.
Answer
To find the derivative ( f'(x) ), we can use the quotient rule. If we have ( f(x) = \frac{u(x)}{v(x)} ) where ( u(x) = 4x - 6 ) and ( v(x) = 2x - 1 ), then:
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Differentiate:
Using the quotient rule, ( f'(x) = \frac{u'v - uv'}{v^2} ) [ u' = 4, \quad v' = 2 ] -
Substitute:
[ f'(x) = \frac{(4)(2x - 1) - (4x - 6)(2)}{(2x - 1)^2} ] [ = \frac{(8x - 4) - (8x - 12)}{(2x - 1)^2} ] -
Simplify:
Combining terms:
[ f'(x) = \frac{8}{(2x - 1)^2} ]
Thus, the derivative in its simplest form is ( f'(x) = \frac{8}{(2x - 1)^2} ).