Final answer:
To solve the derivative of the function 4 log_5(4x + 5), we apply the logarithmic differentiation rule and chain rule to obtain the derivative, which simplifies to 16/((4x + 5)ln(5)). Option fourth is the correct answer.
Step-by-step explanation:
We are solving the derivative of the function 4 log5(4x + 5). The logarithm rule to apply here is the derivative of logb(x), which is 1/(xln(b)) when base b is a constant. Additionally, according to the chain rule, we need to multiply by the derivative of (4x + 5), which is 4. Therefore, the derivative of our function will be:
4 * d/dx[log5(4x + 5)] = 4 * (1/((4x + 5)ln(5)) * d/dx[4x + 5])
This simplifies to:
4 * (4/(4x + 5)ln(5))
Reducing the 4s gives us:
16/((4x + 5)ln(5))
This appears to match one of the provided options, confirming that the correct answer is:
StartFraction 16 Over (4 x + 5) (ln 5) EndFraction.