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derivative of 10^x

derivative of 10^x

2 min read 29-03-2025
derivative of 10^x

The derivative of 10x 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 nxn-1. However, this rule applies when the base is a variable and the exponent is a constant. In 10x, 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 = 10x. Taking the natural logarithm (ln) of both sides gives: ln(y) = ln(10x)

  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 = 10x, we can substitute this back into the equation: dy/dx = 10x ln(10)

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

Method 2: Using the Definition of the Exponential Function

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

  1. Rewrite using base e: Remember that ab = eb ln(a). Applying this, we get 10x = ex ln(10)

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

  3. Simplify: Substitute back 10x for ex ln(10): The derivative is 10x ln(10)

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

Example: Finding the Derivative at a Specific Point

Let's find the derivative of 10x 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 ex.

Conclusion

The derivative of 10x is 10x 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.

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