Mastering Derivatives of Logarithmic and Exponential Functions

Understanding how to differentiate logarithmic and exponential functions is a fundamental skill in calculus. These functions have wide-ranging applications in various fields, including physics, engineering, and economics. In this guide, we will explore the techniques for finding the derivatives of these functions using integration by parts, the change of base formula, and general derivative rules.

Integration by Parts for Logarithmic Functions

Consider the following integral:

I ∫ ln x dx

To evaluate this, we use the method of integration by parts. We start by identifying u and v as follows:

u ln x dv dx

Thus, we have:

du (1/x) dx v x

Applying the formula for integration by parts:

∫ u dv uv - ∫ v du

We get:

I x ln x - ∫ x (1/x) dx

This simplifies to:

I x ln x - ∫ dx x ln x - x C

Qualitative Properties of Logarithmic Functions

Logarithmic functions exhibit several qualitative properties that are useful in differentiation. These include:

loga(bc) logab logac loga(bc) c logab

Let's consider the function f(x) log14(14x8) - log2(23x).

Using the properties of logarithms, we can rewrite this as:

f(x) log14(14) 8 log14 x - 3x log2(2)

Simplifying further:

f(x) 1 8 log14 x - 3x

The derivative of this function is:

f'(x) 8/x loge 14 - 3

Change of Base Formula

The change of base formula is a powerful tool for rewriting logarithmic functions in terms of natural logarithms. This is particularly useful when dealing with more complex expressions. For instance:

f(x) log7(14x8) 23x

We can rewrite this as:

f(x) ln(14x8) / ln 7 23x

This can further be simplified to:

f(x) (ln 14 8 ln x) / ln 7 23x

General Logarithmic and Exponential Derivatives

Let's consider the general forms:

fx logba xc g(x) abx

We know that:

d/dx log x 1/x. Here, log ≡ loge ≡ ln d/dx ex ex

The derivative of a general logarithmic function is:

logba ≡ (logca / logcb)

Thus, for fx logba xc (log xc / log b) (c log x / log b):

f'(x) c / (x log b)

For a general exponential function, g(x) abx, we can rewrite it as:

abx (ebx log a)

Thus, taking the derivative:

g'(x) b ln a ebx log a b ln a g(x)

These techniques will help you tackle a wide range of differentiation problems involving logarithmic and exponential functions. Remember, understanding the properties and rules of these functions can significantly simplify the process.