Question

Differentiate with respect to x
Y=log5x using chain rule

Answer 1

To differentiate \( Y = \log_5 x \) with respect to \( x \) using the chain rule, we first need to express the logarithm in terms of a more familiar base, such as the natural logarithm (ln).

We can use the change of base formula for logarithms:

\[
Y = \log_5 x = \frac{\ln x}{\ln 5}
\]

Now, we can differentiate \( Y \) with respect to \( x \). Since \( \ln 5 \) is a constant, we can factor it out of the derivative:

\[
\frac{dY}{dx} = \frac{1}{\ln 5} \cdot \frac{d}{dx}(\ln x)
\]

The derivative of \( \ln x \) with respect to \( x \) is \( \frac{1}{x} \). Therefore, we have:

\[
\frac{dY}{dx} = \frac{1}{\ln 5} \cdot \frac{1}{x}
\]

Combining these results, the derivative of \( Y = \log_5 x \) with respect to \( x \) is:

\[
\frac{dY}{dx} = \frac{1}{x \ln 5}
\]