derivative of 10^x

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29-03-2025

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The derivative of 10^x might seem daunting at first, but it's easily solved using logarithmic differentiation and understanding the rules of calculus. This guide will walk you through the process step-by-step, explaining the underlying principles and providing examples.

Understanding the Challenge: Why the Power Rule Doesn't Apply Directly

You might initially think you can apply the power rule of differentiation, which states that the derivative of xⁿ is nx^n-1. However, this rule applies when the base is a variable and the exponent is a constant. In 10^x, the base is a constant (10) and the exponent is a variable (x). Therefore, a different approach is needed.

Method 1: Logarithmic Differentiation

Logarithmic differentiation provides an elegant solution. Here's how it works:

1. Take the natural logarithm of both sides: Let y = 10^x. Taking the natural logarithm (ln) of both sides gives: ln(y) = ln(10^x)

2. Use the power rule of logarithms: This simplifies the equation to: ln(y) = x ln(10)

3. Differentiate both sides implicitly with respect to x: Remember that the derivative of ln(y) is (1/y) * (dy/dx). Applying this, and remembering that ln(10) is a constant, we get: (1/y) * (dy/dx) = ln(10)

4. Solve for dy/dx: Multiply both sides by y to isolate the derivative: dy/dx = y * ln(10)

5. Substitute back for y: Since y = 10^x, we can substitute this back into the equation: dy/dx = 10^x ln(10)

Therefore, the derivative of 10^x is 10^x ln(10).

Method 2: Using the Definition of the Exponential Function

The function 10^x can be rewritten using the exponential function with base e:

1. Rewrite using base e: Remember that a^b = e^{b ln(a)}. Applying this, we get 10^x = e^{x ln(10)}

2. Differentiate using the chain rule: The derivative of e^u is e^u * du/dx. Applying the chain rule with u = x ln(10), we get: d/dx (e^{x ln(10)}) = e^{x ln(10)} * ln(10)

3. Simplify: Substitute back 10^x for e^{x ln(10)}: The derivative is 10^x ln(10)

Again, we arrive at the same result: the derivative of 10^x is 10^x ln(10).

Example: Finding the Derivative at a Specific Point

Let's find the derivative of 10^x at x = 2.

1. Substitute x = 2 into the derivative: dy/dx = 10² ln(10)

2. Calculate: This simplifies to 100 ln(10) ≈ 230.26

Why is ln(10) Involved?

The presence of ln(10) is a consequence of changing the base of the exponential function from 10 to the natural base e. The natural logarithm allows us to work with the more readily differentiable exponential function e^x.

Conclusion

The derivative of 10^x is 10^x ln(10). This result is obtained through logarithmic differentiation or by rewriting the function using the natural exponential function and applying the chain rule. Understanding this process provides a solid foundation for tackling more complex derivative problems. Remember that the key is recognizing when the standard power rule is inapplicable and employing the appropriate techniques, like logarithmic differentiation.