The derivative of (10^x) is (10^x \ln 10), where (\ln) denotes the natural logarithm (base (e)). This post explains how to compute the derivative of (10) raised to (x) using logarithmic differentiation.
Derivative of (10^x): Step-by-Step Explanation
Key Formula
[ \frac{d}{dx}(10^x) = 10^x \ln 10 ]
Proof Using Logarithmic Differentiation
- Let (y = 10^x).
- Take the natural logarithm (ln) of both sides:
[ \ln y = \ln(10^x) ]
[ \ln y = x \ln 10 ] - Differentiate both sides with respect to (x):
[ \frac{1}{y} \frac{dy}{dx} = \ln 10 ] - Solve for (\frac{dy}{dx}):
[ \frac{dy}{dx} = y \ln 10 ]
[ \frac{dy}{dx} = 10^x \ln 10 ]
Thus, the derivative of (10^x) is (10^x \ln 10).
Example: Derivative at (x = 0)
[ \left.\frac{d}{dx}(10^x)\right|_{x=0} = 10^0 \ln 10 = 1 \cdot \ln 10 = \ln 10 ]
Frequently Asked Questions (FAQ)
Q1: What is the derivative of (10^x)?
A: The derivative is (10^x \ln 10).
Q2: How is the derivative of (10^x) derived?
A: It is derived using logarithmic differentiation, as shown above.
Q3: What is the derivative of (10^x) at (x = 0)?
A: At (x = 0), the derivative equals (\ln 10).
Q4: Does the derivative formula apply to other exponential functions like (2^x)?
A: Yes! The general rule is:
[ \frac{d}{dx}(a^x) = a^x \ln a ]
For example, (\frac{d}{dx}(2^x) = 2^x \ln 2).
Related Topics
👉 Derivative of Exponential Functions
👉 Logarithmic Differentiation Explained
By understanding this derivation, you can apply similar methods to other exponential functions. For deeper insights, check out advanced calculus resources or explore more derivatives of exponential and logarithmic functions.
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