Question

In Exercises \(1-28\) , find \(d y / d x\) . Remember that you can use NDER to support your computations. $$y=\log _{c} \sqrt{x}$$

Short answer

The derivative of the given function with respect to \( x \), \( dy/dx \), is \( 1/(2x) \)

Step-by-step solution

Rewrite the square root as a power

According to properties of roots, we can rewrite \( \sqrt{x} \) as \( x^{1/2} \). Therefore, the function becomes \( y = \log_c (x^{1/2}) \).

Use the properties of logarithms

According to the properties of logarithms, we can bring down the exponent in front. The function becomes \( y = (1/2) \log_c (x) \).

Differentiate using chain rule

Using the chain rule, the derivative of \( y = a \log_c(bx) \) is \( (a/b) \cdot 1/x \) (assuming that \( c \) is just a constant). Therefore, the derivative comes out to be \( (1/2) \cdot 1/x = 1/(2x) \) .