Use logarithmic differentiation

Question

y=(x³+2)²(x⁵+4)⁴

Answer 1

First take the log on both sides,

lny = ln[(x³+2)²(x⁵+4)⁴]

Using logarithm properties, we can simplify.

lny = 2ln(x³+2) + 4ln(x⁵+4)

Differentiate on both sides.

(1/y)(dy/dx) = (2/x³+2)*(3x²) + (4/x⁵+4)(5x⁴)

Multiply by y on both sides

dy/dx = y[(2/x³+2)*(3x²) + (4/x⁵+4)(5x⁴)]

Now, you the substitute the value of y

dy/dx = [(x³+2)²(x⁵+4)⁴]*[(2/x³+2)*(3x²) + (4/x⁵+4)(5x⁴)]

I hope this helps. If you have any question, please let me know in the comments.