find the derivative

Question

  Find the derivative.  

                ______

#3. y=ln √x^2+5

 

#5. y=ln(5x^3-6x)-9x

 

#7. y=4e^6x^2-4

 

#8. y=ln(3x+5)

 

#10.y=1-9e^-x^3

 

#18. f(x)=x^-7-5e^x

Answer 1

If y = ln(u(x)), then y' = du/dx / u.

 

 

If y = e^u(x), then y' = e^u(x) du/dx.

 

 

3. y=ln√(x² + 5) = ½ln(x² + 5)

 

 

y' = ½(2x/(x² + 5)) = x/(x² + 5)

 

 

 

5. y=ln(5x³ - 6x) - 9x

 

y' = (15x² - 6)/(5x³ - 6x) - 9

 

 

7. y = 4e^{6x^2}- 4

 

y' = 4e^{6x^2}(12x) = 48xe^{6x^2}

 

 

 
8. y=ln(3x+5)

 

y' = 3/(3x + 5)

 

 

 
10. y = 1 - 9e^{-x^3}

 

y' = -9e^{-x^3}(-3x²) = 27x²e^{-x^3}

 

 

 

18. f(x) = x^-7 - 5e^x

 

f'(x) = -7x^-8 - 5e^x