Asked by Michael
                What are the derivatives?  Thanks
a) y = log(base2)(4-9x)
b) y = x^(8x)
            
        a) y = log(base2)(4-9x)
b) y = x^(8x)
Answers
                    Answered by
            Reiny
            
    in general for y = log<sub>a</sub> u
dy/dx = (1/ln a)(1/u)du/dx
so for the above
dy/dx = (1/ln2)(-9)/(4-9x)
for the second one take ln of both sides
y = x^(8x)
ln y = ln (x^(8x))
ln y = (8x)(ln x)
now take the derivative implicityly using the product rule on the right side
y'/y = 8x(1/x) + 8(ln x)
y' = y[8 + 8ln x)
= (8+8ln x)(x^(8x))
    
dy/dx = (1/ln a)(1/u)du/dx
so for the above
dy/dx = (1/ln2)(-9)/(4-9x)
for the second one take ln of both sides
y = x^(8x)
ln y = ln (x^(8x))
ln y = (8x)(ln x)
now take the derivative implicityly using the product rule on the right side
y'/y = 8x(1/x) + 8(ln x)
y' = y[8 + 8ln x)
= (8+8ln x)(x^(8x))
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