Find the derivative of the given expression. f(x) = ln((21x + 8)^4) f'(x) =

To find the derivative of the function f ( x ) = ln (( 21 x + 8 ) 4 ) , we can use a combination of the chain rule and the properties of logarithms.
Step-by-step Solution:

Recognize the structure of the function:
The given function f ( x ) = ln (( 21 x + 8 ) 4 ) is a logarithm of an expression raised to a power.

Use the properties of logarithms to simplify:
We can simplify ln (( 21 x + 8 ) 4 ) using the power rule of logarithms. This property states that ln ( a b ) = b ⋅ ln ( a ) . So, our function becomes:
f ( x ) = 4 ⋅ ln ( 21 x + 8 )

Differentiate using the chain rule:
The derivative of ln ( u ) with respect to u is u 1 ​ . Here, u = 21 x + 8 , so we also need to multiply by the derivative of u with respect to x , which is 21 .
( f'(x) = 4 \cdot \frac{1}{21x + 8} \cdot 21 )

Simplify the derivative:
( f'(x) = \frac{84}{21x + 8} )


So, the derivative of the function f ( x ) = ln (( 21 x + 8 ) 4 ) is f ′ ( x ) = 21 x + 8 84 ​ . This result tells us the rate of change of the function with respect to x . The process involves differentiating step-by-step while applying logarithmic differentiation rules.

Answered by EmmaGraceJohnson | 2025-07-21