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Question
Differentiate the following:
y = (2x – 5)4 (8x2 – 5)–3
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Solution
y = (2x – 5)4 (8x2 – 5)–3
`("d"y)/("d"x) = (2x - 5)^4 xx "d"/("d"x) (8x^2 - 5)^(-3) + (8x^2 - 5)^(-3) xx "d"/("d"x) (2x - 5)^4` .......(By product rule)
`("d"y)/("d"x) = (2x - 5)^4 xx - 3(8x^2 - 5)^(- 3 - 1) (8 xx 2x - 0) + (8x^2 - 5)^(-3) xx 4(2x - 5)^(4 - 1) (2 xx 1 - 0)`
`("d"y)/("d"x) = - 3(2x - 5)^4 (8x^2 - 5)^(-4) xx 16x + (8x^2 - 5)^(-3) xx 4(2x - 5)^3 (2 xx 1)`
`("d"y)/("d"x) = - 48x (2x - 5)^4 (8x^2 - 5)^(- 4) + 8(8x^2 - 5)^(- 3) (2x - 5)^3`
= `-48x (2x - 5)^4 xx 1/(8x^2 - 5)^4 + (8(2x - 5)^3)/(8x^2 - 5)^3`
= `(8(2x - 5)^3)/(8x^2 - 5)^3 [1 - (6x(2x - 5))/(8x^2 - 5)]`
= `(8(2x - 5)^3)/(8x^2 - 5)^3 xx [(8x^2 - 5 - 12x^2 + 30x)/(8x^2 - 5)]`
= `(8(2x - 5)^3)/(8x^2 - 5)^3 xx [- 4x^2 + 30x - 5]`
`("d"y)/("d"x) = (8(2x - 5)^3)/(8x^2 - 5)^4 xx (30x - 4x^2 - 5)`
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