Math Problem Statement

What is the derivative of 10^x with respect to x?

Solution

To solve the derivative of 10x10^x, we can use the general rule for differentiating exponential functions of the form axa^x, which is:

ddx(ax)=axln(a)\frac{d}{dx}(a^x) = a^x \ln(a)

Here, a=10a = 10. Therefore, the derivative of 10x10^x is:

ddx(10x)=10xln(10)\frac{d}{dx}(10^x) = 10^x \ln(10)

So, the correct answer from the given options is D: 10xln(10)10^x \ln(10).

Would you like more details on how to differentiate exponential functions or have any further questions?

Here are some related questions:

  1. How do you differentiate general exponential functions like axa^x?
  2. What is the derivative of exe^x and how does it compare to axa^x?
  3. Can you explain how logarithms come into play when differentiating exponentials?
  4. How would you apply the chain rule to derivatives of exponential functions like 10f(x)10^{f(x)}?
  5. How can you differentiate expressions involving both exponents and logs, like ln(10x)\ln(10^x)?

Tip: Remember that for bases other than ee, the natural logarithm ln(a)\ln(a) appears in the derivative!

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Differentiation

Formulas

d/dx (a^x) = a^x * ln(a)

Theorems

Derivative of Exponential Functions

Suitable Grade Level

Grades 11-12