Use logarithmic differentiation to find the derivative of the function. y = x^8^sin x

Question

Please help me

Answer 1

I assume that the function is supposed to be:  y = (x⁸)^{sin x}.  Is that correct?

 

If so, then what I would do is start by taking the log of both sides:

 

ln y = sin x ln x⁸ = (sin x) * (8 ln x) = 8 (sin x)(ln x)

 

Now differentiate both sides.  On the left you'll need to use the chain rule, and on the right you'll use the product rule:

 

1/y dy/dx = 8[(sin x) (1/x) + (cos x)(ln x)] = 8 [(sin x) / x + (cos x)(ln x)]

 

Multiply both sides by y:\

 

dy/dx = y * 8 [(sin x) / x + (cos x)(ln x)]

 

Since y = (x⁸)^{sin x}, we can rewrite this as:

 

dy/dx = (x⁸)^{sin x} * 8 [(sin x) / x + (cos x)(ln x)] = 8(x⁸)^{sin x}  [(sin x) / x + (cos x)(ln x)]