Question

Differentiate and simplify please help ??

Answer 1

7.a) `(dy)/dx=1/log2(4/x-3cotx)`

7.b) `(dy)/dx=1/log3(log3+1/(x-2))`

Explanation:

7.a)
Given:
`f(x)=log_2|(x^4)/sin^3(x)|`
By change of base rule,

`log_2|(x^4)/sin^3(x)|=(log|(x^4)/sin^3(x)|)/log2=1/log2(logx^4-logsin^3x)`
By index rule
`=1/log2(4logx-3logsinx)`
Differentiating wrt x
`(dy)/dx=1/log2(4/x-3/sinxcosx)`

`(dy)/dx=1/log2(4/x-3cotx)`

7.b)
Given:
`f(x)=log_3|3^x(x-2)|`
Change of base rule
`log_3|3^x(x-2)|=(log|(3^x(x-2)|))/log3=1/log3(log3^x+log(x-2))`
By index rule
`=1/log3(xlog3+log(x-2))`
Differentiating wrt x
`(dy)/dx=1/log3(log3+1/(x-2))`