Question

Calculate the derivative of `y = (x^2+2)^2(x^4+4)^4` using logarithms?

Answer 1

`dy/dx = (4x) ( 5x^4 + 8x^2 + 4 ) (x^2+2)(x^4+4)^3`

Explanation:

`y = (x^2+2)^2(x^4+4)^4`

Taking Natural Logarithms:

`ln y = ln {(x^2+2)^2(x^4+4)^4}`
`:. ln y = ln (x^2+2)^2 + ln (x^4+4)^4`
`:. ln y = 2ln (x^2+2) + 4ln (x^4+4)`

Differentiating we get:
`1/ydy/dx = 2 (2x)/(x^2+2) + 4 (4x^3)/(x^4+4)`

Simplifying;
`1/ydy/dx = (4x) { 1/(x^2+2) + (4x^2)/(x^4+4) }`
`:. 1/ydy/dx = (4x) { ( (x^4+4) + 4x^2(x^2+2) )/( (x^2+2)(x^4+4) ) }`
`:. 1/ydy/dx = (4x) { ( x^4+4 + 4x^4 + 8x^2 )/( (x^2+2)(x^4+4) ) }`
`:. 1/ydy/dx = (4x) { ( 5x^4 + 8x^2 + 4 )/( (x^2+2)(x^4+4) ) }`
`:. dy/dx = (4x) { ( 5x^4 + 8x^2 + 4 )/( (x^2+2)(x^4+4) ) } y`
`:. dy/dx = (4x) { ( 5x^4 + 8x^2 + 4 )/( (x^2+2)(x^4+4) ) } (x^2+2)^2(x^4+4)^4`
`:. dy/dx = (4x) ( 5x^4 + 8x^2 + 4 ) (x^2+2)(x^4+4)^3`