Learn how to find the derivative of 3^x. The derivative is d/dx[3^x].
Below is the graph of f(x) = 3^x and its derivative f'(x). Notice how the original function and derivative relate to each other.
Original Function:
Derivative:
This derivative represents the rate of change of the original function. The calculation follows standard differentiation rules as shown in the step-by-step solution below.
The Power Rule applies to functions where a constant base is raised to a variable exponent. In this case, the function 3^x fits this structure because the base (3) is constant, and the exponent (x) is the variable.
💡 Why this works: To differentiate 3^x, we recognize that this is an exponential function with a constant base. The Power Rule for exponential functions tells us how to find the derivative of such functions.
We apply the Power Rule, which for exponential functions of the form a^x, states that the derivative is a^x * ln(a).
💡 Why this works: For 3^x, we differentiate it by multiplying 3^x by the natural logarithm of 3 (ln(3)). This gives the derivative: d/dx[3^x] = 3^x * ln(3).
Once we apply the rule, the derivative of 3^x simplifies to 3^x * ln(3), which represents the rate of change of the function.
💡 Why this works: We have the final form of the derivative, d/dx[3^x] = 3^x * ln(3). This is the correct derivative because it follows from the Power Rule for exponential functions and is consistent with the general form of such derivatives.
[Teaching explanation - to be filled]
[Application to this function - to be filled]
❌ Common Mistake:
Not simplifying the final answer
✅ Correct Approach:
Always simplify your derivative to its most reduced form
Double-check your work by comparing your steps with our solution. Use our calculator to verify each step of the differentiation process.
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