Use logarithmic differentiation: (x^2 + 2)^2 * (x^4 + 4^4

Question

Use logarithmic differentiation: (x² + 2)² * (x⁴ + 4)⁴

 

 

ANSWER: (x² + 2)² * (x ⁴+ 4)⁴ * [4x / (x² + 2)  +  16x³ / (x⁴ + 4)]

#39, p223

Answer 1

y = (x²+2)²+(x⁴+4)⁴

 

The idea is to take the natural log of both sides, rewrite the right side using properties of logarithms and then take the derivative of both sides with respect to x.

 

ln y = 2ln(x²+2) + 4ln(x⁴+4)

 

Now the derivative of both sides:

(1/y) y' = (2)[1/(x²+2)](2x) + 4[1/(x⁴+4)](4x³) : the factors 2x and 4x³ appear because of the chain rule.

 

Now multiply both sides by y to isolate y'.

 

y' = (the original value for y in terms of x) * (simplified version of the terms shown above. 

 

The answer from #39 on page 223 is correct.

 

Note that logarithmic differentiation is used in lieu of the product rule or the quotient rule when the function can be deemed "complicated".