# Derivative of 3^x Explained

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147 # Derivative of 3^x Explained

The derivative of 3^x is 3^x * ln(3). To see this, you can use the power rule for derivatives: if f(x) = x^n, then f'(x) = n * x^(n-1).

So, if you let f(x) = 3^x, then f'(x) = x * 3^(x-1). But 3^(x-1) is just another way of writing 3^x, so f'(x) = x * 3^x. And since ln(3) is just the exponent to which you must raise 3 to get 3^x, we can rewrite this as 3^x * ln(3).

The power rule for derivatives states that if f(x) = x^n, then the derivative of f(x), denoted as f'(x), is given by f'(x) = n * x^(n-1). This is a general formula that you can use to find the derivative of any function of the form f(x) = x^n, where n is a constant.

For example, suppose you want to find the derivative of f(x) = x^2. Using the power rule, we have:

f'(x) = 2 * x^(2-1) = 2 * x^1 = 2x

So the derivative of f(x) = x^2 is f'(x) = 2x.

Now, let’s consider the function g(x) = 3^x. We can use the power rule to find the derivative of this function as follows:

g'(x) = x * 3^(x-1)

But 3^(x-1) is just another way of writing 3^x, so we can rewrite this as:

g'(x) = x * 3^x

Finally, since ln(3) is the exponent to which you must raise 3 to get 3^x (that is, 3^ln(3) = 3^x), we can rewrite g'(x) as:

g'(x) = 3^x * ln(3)

So the derivative of g(x) = 3^x is g'(x) = 3^x * ln(3).

## How do you differentiate 3^x with respect to x?

To differentiate 3^x with respect to x, you can use the power rule for derivatives. The power rule states that if f(x) = x^n, then the derivative of f(x), denoted as f'(x), is given by f'(x) = n * x^(n-1).

Applying the power rule to the function f(x) = 3^x, we have:

f'(x) = x * 3^(x-1)

But 3^(x-1) is just another way of writing 3^x, so we can rewrite this as:

f'(x) = x * 3^x

Finally, since ln(3) is the exponent to which you must raise 3 to get 3^x (that is, 3^ln(3) = 3^x), we can rewrite f'(x) as:

f'(x) = 3^x * ln(3)

So the derivative of f(x) = 3^x with respect to x is f'(x) = 3^x * ln(3).

## Derivative of 3/x using limit definition

To find the derivative of the function f(x) = 3/x using the limit definition, we can use the following formula:

f'(x) = lim(h->0) [(3/(x+h)) – (3/x)] / h

This formula states that the derivative of f(x) at a point x is equal to the limit of the difference quotient of f(x) as h approaches 0.

Substituting 3/x for f(x) and simplifying, we get:

f'(x) = lim(h->0) [(3/(x+h)) – (3/x)] / h = lim(h->0) (3/x – 3/(x+h)) / h = lim(h->0) (3x – 3(x+h)) / (xh(x+h)) = lim(h->0) (3x – 3x – 3h) / (xh(x+h)) = lim(h->0) (-3h) / (xh(x+h)) = lim(h->0) -3 / (x(x+h)) = -3 / x^2

Therefore, the derivative of f(x) = 3/x is f'(x) = -3/x^2.