Question

In Exercises \(1-28\) , find \(d y / d x\) . Remember that you can use NDER to support your computations. $$y=\log _{10} e^{x}$$

Short answer

The derivative of function \(y=\log _{10} e^{x}\) is \(dy/dx = 1/ln(10)\)

Step-by-step solution

Rewrite the Function using Change of Base Formula

Any base logarithm can be represented using natural logarithm. The change of base formula states that \(log_b{a} = ln(a)/ln(b)\). Therefore, rewrite the function as \(y = ln{e^x} / ln{10}\)

Simplify the Function

By properties of natural logarithms, we know that \(ln(e^x) = x\). Therefore, our function simplifies to \(y = x / ln(10)\)

Compute the Derivative dy/dx

The function in this step is a simple linear function, so to find its derivative we apply the constant rule of differentiation, which states that the derivative of a constant times a function is just the constant times the derivative of the function. So the derivative of y, denoted as \(dy/dx\), is \(1/ln(10)\), because the derivative of x is 1.