Derivatives of Exponential and Logarithmic Functions

Today, we will start with the derivatives of exponential functions. Let's begin with the natural exponential function, which is $e^x$. Does anyone know what the derivative of $e^x$ is?

I think it's just $e^x$, right?

Exactly! The derivative of $e^x$ is indeed $e^x$. This means that the rate at which $e^x$ changes is proportional to its current value, which is a unique property of the exponential function.

What about other bases? Is there a formula for that?

Great question! For an exponential function with a different base, say $a^x$, the derivative is given by $\frac{d}{dx}[a^x] = a^x \ln(a)$. So, if you know the base, you can easily compute the derivative!

Can you give an example with a specific base?

Sure! Let’s take $f(x) = 2^x$. The derivative would be $\frac{d}{dx}[2^x] = 2^x \ln(2)$.

So, if we substitute a value like $x = 1$, would it help us find the rate of change?

Exactly! If you plug in $x = 1$, you'd get $2^1 \ln(2)$, which gives you the instantaneous rate of change at that point.

Now, let’s transition to logarithmic functions. Who can tell me the derivative of $\ln(x)$?

I believe it's $\frac{1}{x}$.

Correct! The derivative of $\ln(x)$ is $\frac{1}{x}$. This explains how the natural logarithm grows slower than polynomial functions.

What about logarithms with different bases?

For logarithms to any base, such as $\log_a(x)$, the derivative is given by $\frac{1}{x \, \ln(a)}$. This means that the growth rate depends on both the input value and the logarithmic base.

Could you show a practical application for finding the derivative of a logarithm?

Certainly! In real-life situations, such as calculating pH in chemistry, we often use logarithmic functions where understanding the rate of change is crucial. For instance, if we're dealing with $\log_{10}(x)$, we'd apply the formula $\frac{1}{x \, ext{ln}(10)}$.