Calculating Derivatives of Logarithmic Functions Using the Chain Rule | Step-by-Step Guide

dy/dx log_(c) (u)

To find the derivative of the function y = log_c(u) with respect to x, we will use the chain rule

To find the derivative of the function y = log_c(u) with respect to x, we will use the chain rule.

Let’s break down the problem step by step:

Step 1: Rewrite y = log_c(u) using logarithmic properties.
– Recall that log_c(u) can be written as ln(u)/ln(c), where ln represents the natural logarithm.
– So, y = ln(u)/ln(c).

Step 2: Apply the chain rule.
– The chain rule states that if f(g(x)) is a composite function, then its derivative is given by f'(g(x)) multiplied by g'(x).

In our case, the outer function is y = ln(u)/ln(c) and the inner function is u(x).

Derivative of the outer function:
– To find the derivative of ln(u)/ln(c) with respect to u, treat ln(c) as a constant and take the derivative of ln(u) using the chain rule.
– The derivative of ln(u) with respect to u is 1/u.
– Since ln(c) is a constant, its derivative is zero.

Derivative of the inner function:
– To find the derivative of u(x) with respect to x, we need additional information about the function u(x). Please provide the expression for u(x), and we can proceed from there.

Once we have the derivative of the inner function, we can multiply it with the derivative of the outer function to get dy/dx.

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