Differentiate?

Question

y = ln(x^7sin^2x)

help?

Answer 1

y = ln(x^7 sin^2(x))

One option is to differentiate directly using the chain rule followed by the product rule. I think it's better, however, to apply log identities prior to differentiating. In particular, the identity I speak of is that the log of a product is the sum of logs.

y = ln(x^7) + ln(sin^2(x))

Now, we can move the exponents to the outside.

y = 7 ln(x) + 2 ln(sin(x))

Differentiating,

dy/dx = 7(1/x) + 2(1/sin(x))(cos(x))
dy/dx = (7/x) + 2(cos(x)/sin(x))
dy/dx = (7/x) + 2cot(x)

Answer 2

Let u = (x^7) (sin ² x)
du / dx = 7 (x^6) (sin ² x) + 2 sin x cos x (x^7)
du / dx = x^6 sin x (7 sin x + 2 x cos x)
y = ln u
dy / du = 1 / u
dy / du = 1 / (x^7 sin ² x)
dy / dx = (dy / du ) (du / dx)
dy / dx
= [1 / (x^7 sin ² x) ] [x^6 sin x (7sin x + 2x cos x) ]
= (1 / x sin x) (7 sin x + 2x cos x)
= (1 / x) (7 + 2x cot x)