Differentiate.
[tex]f(x)=\ln \left[\frac{(2 x+9)(x+7)^6}{(1-6 x)^3}\right][/tex]
[tex]\frac{d}{d x}\left[\ln \left[\frac{(2 x+9)(x+7)^6}{(1-6 x)^3}\right]\right]=[/tex]
Answer (1)
Expand the logarithmic function using properties of logarithms: ln ( ab ) = ln a + ln b , ln ( a / b ) = ln a − ln b , and ln a n = n ln a .
Differentiate each term using the chain rule and the derivative of ln x , which is x 1 .
Combine the derivatives and simplify the expression by finding a common denominator.
The derivative of the given function is: 12 x 3 + 136 x 2 + 355 x − 63 2 ( 24 x 2 − 10 x − 601 ) .
Explanation
Problem Setup We are asked to find the derivative of the function
f ( x ) = ln [ ( 1 − 6 x ) 3 ( 2 x + 9 ) ( x + 7 ) 6 ]
To do this, we'll use properties of logarithms to expand the function and then apply differentiation rules.
Expanding the Function First, we expand the function using logarithm properties:
ln ( ab ) = ln a + ln b
ln ( a / b ) = ln a − ln b
ln a n = n ln a
Applying these properties, we get:
f ( x ) = ln ( 2 x + 9 ) + ln (( x + 7 ) 6 ) − ln (( 1 − 6 x ) 3 )
f ( x ) = ln ( 2 x + 9 ) + 6 ln ( x + 7 ) − 3 ln ( 1 − 6 x )
Differentiating Each Term Now, we differentiate each term with respect to x :
d x d ln ( 2 x + 9 ) = 2 x + 9 2
d x d 6 ln ( x + 7 ) = x + 7 6
d x d − 3 ln ( 1 − 6 x ) = − 3 ⋅ 1 − 6 x − 6 = 1 − 6 x 18
Combining Derivatives Combining these derivatives, we get:
d x d f ( x ) = 2 x + 9 2 + x + 7 6 + 1 − 6 x 18
Simplifying the Expression To simplify the expression, we find a common denominator:
2 x + 9 2 + x + 7 6 + 1 − 6 x 18 = ( 2 x + 9 ) ( x + 7 ) ( 1 − 6 x ) 2 ( x + 7 ) ( 1 − 6 x ) + 6 ( 2 x + 9 ) ( 1 − 6 x ) + 18 ( 2 x + 9 ) ( x + 7 )
Expanding the numerator:
2 ( x + 7 ) ( 1 − 6 x ) = 2 ( x − 6 x 2 + 7 − 42 x ) = 2 ( − 6 x 2 − 41 x + 7 ) = − 12 x 2 − 82 x + 14
6 ( 2 x + 9 ) ( 1 − 6 x ) = 6 ( 2 x − 12 x 2 + 9 − 54 x ) = 6 ( − 12 x 2 − 52 x + 9 ) = − 72 x 2 − 312 x + 54
18 ( 2 x + 9 ) ( x + 7 ) = 18 ( 2 x 2 + 14 x + 9 x + 63 ) = 18 ( 2 x 2 + 23 x + 63 ) = 36 x 2 + 414 x + 1134
Adding these together:
( − 12 x 2 − 82 x + 14 ) + ( − 72 x 2 − 312 x + 54 ) + ( 36 x 2 + 414 x + 1134 ) = ( − 12 − 72 + 36 ) x 2 + ( − 82 − 312 + 414 ) x + ( 14 + 54 + 1134 ) = − 48 x 2 + 20 x + 1202
Expanding the denominator:
( 2 x + 9 ) ( x + 7 ) ( 1 − 6 x ) = ( 2 x 2 + 14 x + 9 x + 63 ) ( 1 − 6 x ) = ( 2 x 2 + 23 x + 63 ) ( 1 − 6 x ) = 2 x 2 + 23 x + 63 − 12 x 3 − 138 x 2 − 378 x = − 12 x 3 − 136 x 2 − 355 x + 63
So, the derivative is:
− 12 x 3 − 136 x 2 − 355 x + 63 − 48 x 2 + 20 x + 1202 = 12 x 3 + 136 x 2 + 355 x − 63 2 ( 24 x 2 − 10 x − 601 )
Final Answer Therefore, the derivative of the given function is:
d x d [ ln [ ( 1 − 6 x ) 3 ( 2 x + 9 ) ( x + 7 ) 6 ] ] = 12 x 3 + 136 x 2 + 355 x − 63 2 ( 24 x 2 − 10 x − 601 )
Examples
In chemical kinetics, you might encounter a rate law expression that involves a product of concentrations raised to certain powers, all within a logarithmic function. Differentiating such an expression helps you determine how the rate of reaction changes with respect to time or concentration. This is crucial for optimizing reaction conditions and understanding reaction mechanisms.
