Find the first derivative

Question

 f(x)=ln(x^3 - 6x^4) g(x)=e^(6x^4) h(x)=(8e^2x)/(x^3)? 

 

 

Answer 1

1)

f(x) = ln(x³ - 6x⁴)

Apply derivative with respect to x both sides.

f'(x) = d/dx [ ln(x³ - 6x⁴) ]

Apply formula : ( d/dx )ln[f(x)] = ln[f(x)] ( d/dx )f(x)

 

= [ 1 / (x³ - 6x⁴) ] (d/dx )(x³ - 6x⁴)

= [ 1 / (x³ - 6x⁴) ] (3x² - 6*4x³)

= [ 1 / (x³ - 6x⁴) ] (3x² - 24x³)

=  (3x² - 24x³) / (x³ - 6x⁴)

=  3x²( 1 - 8x) / x²(x - 6x²)

=  3( 1 - 8x) / (x - 6x²)

f'(x) = 3( 1 - 8x) / (x - 6x²)

Answer 2

(2)

g(x) = e^6x4

Apply derivative with respect to x both sides.

g'(x) = d/dx [ e^6x4 ]

Apply formula : ( d/dx )e^f(x) = e^f(x) ( d/dx )f(x)

= [ e^6x4 ] (d/dx)(6x⁴)

= [e^6x4] (6*4x³)

= [e^6x4] (24x³)

= 24x³e^6x4

g'(x) = 24x³e^6x4

Answer 3

(3)

h(x) = 8e^2x/x³

Apply derivative with respect to x both sides.

h'(x) = d/dx [ 8e^2x/x³ ]

Apply formula : ( d/dx )kf(x) = k( d/dx )f(x)

h'(x) = 8 d/dx [ e^2x/x³ ]

Apply formula : d/dx [ U / V ] = [ Vdu - UdV ] / V²

h'(x) = 8 [ x³(d/dx)(e^2x) - e^2x(d/dx)(x³)] / (x³)²

Apply formula : ( d/dx )e^f(x) = e^f(x) ( d/dx )f(x)

= 8 [ x³e^2x(d/dx)(2x) - e^2x(3x²)] / (x⁶)

= 8 [ 2x³e^2x - 3x² e^2x] / (x⁶)

= 8 x²e^2x[ 2x - 3] / (x⁶)

= 8 e^2x(2x - 3) / x⁴

h'(x) = 8 e^2x(2x - 3) / x⁴