102 LearnersLast updated on 1 September 2025
Derivative of 10^x
We use the derivative of 10^x, which is (10^x) ln(10), as a tool to understand how exponential functions change in response to a small change in x. Derivatives are useful in various real-life applications, such as calculating exponential growth or decay. We will now discuss the derivative of 10^x in detail.
What is the Derivative of 10^x?
The derivative of 10x is commonly represented as d/dx (10x) or (10x)', and its value is (10x) ln(10).
This indicates that the function 10x has a well-defined derivative and is differentiable for all real x.
Key concepts to understand this include:
Exponential Function: 10x is an exponential function with base 10.
Natural Logarithm: ln(10) is the natural logarithm of 10, which is a constant.
Derivative of 10^x Formula
The derivative of 10x can be denoted as d/dx (10x) or (10x)'. The formula for differentiating 10x is: d/dx (10x) = (10x) ln(10) The formula is applicable for all real numbers x.
Proofs of the Derivative of 10^x
We can derive the derivative of 10x using proofs.
To demonstrate this, we use the rules of differentiation and properties of exponential functions.
Methods include:
Using the Exponential Rule
Using the Chain Rule
Using the Exponential Rule
The derivative of 10x can be shown using the exponential rule for differentiation.
Consider the function f(x) = 10x.
The derivative of an exponential function ax is d/dx (ax) = ax ln(a).
Therefore, for f(x) = 10x, the derivative is: f'(x) = 10x ln(10).
Thus, the derivative of 10x is (10x) ln(10).
Using the Chain Rule
To prove the differentiation of 10x using the chain rule,
Consider f(x) = 10x
Express it as f(x) = e(x ln(10))
Using the chain rule, d/dx [e(g(x))] = e(g(x)) g'(x)
Here, g(x) = x ln(10), so g'(x) = ln(10)
The derivative is: d/dx (e(x ln(10))) = e(x ln(10)) ln(10)
Substitute back: f'(x) = 10x ln(10)
Thus, the derivative of 10x is (10x) ln(10).
Higher-Order Derivatives of 10^x
Higher-order derivatives are obtained by differentiating a function multiple times.
For the function 10x, each derivative follows a pattern due to the nature of exponential functions.
First derivative: f′(x) = (10x) ln(10)
Second derivative: f′′(x) = (10x) [ln(10)]2
Third derivative: f′′′(x) = (10x) [ln(10)]3
For the nth derivative of 10x, denoted as fⁿ(x), the formula is: fⁿ(x) = (10x) [ln(10)]ⁿ
Special Cases:
For x = 0, the derivative is (100) ln(10) = ln(10).
For large values of x, the derivative (10x) ln(10) grows rapidly due to the exponential nature.
Common Mistakes and How to Avoid Them in Derivatives of 10^x
Students often make errors when differentiating 10x. Understanding the correct procedures helps prevent these mistakes. Here are some common errors and solutions:
Mistake 1
Incorrect Use of Exponential Rule
Students sometimes apply the wrong rule for the derivative of exponential functions. The correct rule for ax is d/dx (ax) = ax ln(a). Always remember to multiply by the natural logarithm of the base.
Mistake 2
Forgetting the Constant in the Derivative
A common error is neglecting the constant ln(10) in the derivative. Make sure to include ln(10) when differentiating 10x to avoid incorrect results.
Mistake 3
Misunderstanding the Chain Rule
When using the chain rule, students may forget to differentiate the inner function.
For example, differentiating e(x ln(10)) requires applying the chain rule correctly. Ensure each part of the function is differentiated.
Mistake 4
Ignoring the Base of the Exponential Function
Students sometimes confuse 10x with ex, leading to incorrect derivatives. Remember, 10x and ex have different bases, and their derivatives must reflect these differences.
Mistake 5
Not Recognizing Higher-Order Derivatives
Higher-order derivatives require repeated differentiation, which can be challenging. Be consistent in applying differentiation rules for each order of derivative.
Examples Using the Derivative of 10^x
Problem 1
Calculate the derivative of (10^x · x^2)
Here, we have f(x) = 10x · x2.
Using the product rule, f'(x) = u′v + uv′
In the given equation, u = 10x and v = x2.
Let’s differentiate each term, u′ = d/dx (10x) = 10x ln(10) v′ = d/dx (x2) = 2x
Substituting into the given equation, f'(x) = (10x ln(10) · x2) + (10x · 2x)
Let’s simplify terms to get the final answer, f'(x) = 10x x2 ln(10) + 2 · 10x x
Thus, the derivative of the specified function is 10x x2 ln(10) + 2 · 10x x.
Explanation
We find the derivative of the given function by dividing the function into two parts. The first step is finding its derivative and then combining them using the product rule to get the final result.
Problem 2
A company models its revenue growth with the function R(x) = 10^x dollars, where x is the number of years. Find the rate of change of revenue when x = 3 years.
We have R(x) = 10x (revenue function)...(1)
Now, we will differentiate the equation (1)
Take the derivative of 10x: dR/dx = (10x) ln(10)
Given x = 3, substitute this into the derivative: dR/dx = (103) ln(10) dR/dx = 1000 ln(10)
Hence, the rate of change of revenue at x = 3 years is 1000 ln(10) dollars per year.
Explanation
We find the rate of change of revenue at x = 3 years as 1000 ln(10), indicating a rapid increase in revenue due to exponential growth.
Problem 3
Derive the second derivative of the function y = 10^x.
The first step is to find the first derivative, dy/dx = (10x) ln(10)...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [(10x) ln(10)]
Here we use the constant multiple rule, d²y/dx² = ln(10) · d/dx (10x) d²y/dx² = ln(10) · (10x) ln(10) d²y/dx² = (10x) [ln(10)]²
Therefore, the second derivative of the function y = 10x is (10x) [ln(10)]².
Explanation
We use a step-by-step process, starting with the first derivative. Using the constant multiple rule, we differentiate the function again and simplify to find the second derivative.
Problem 4
Prove: d/dx (10^(2x)) = 2 · (10^(2x)) ln(10).
Let’s start using the chain rule:
Consider y = 10(2x)
To differentiate, we use the chain rule: dy/dx = 10(2x) · d/dx (2x) ln(10)
Since d/dx (2x) = 2, dy/dx = 10(2x) · 2 ln(10)
Substituting back, d/dx (10(2x)) = 2 · (10(2x)) ln(10)
Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. We then replace the inner derivative and simplify to derive the equation.
Problem 5
Solve: d/dx (10^x/x)
To differentiate the function, we use the quotient rule:
d/dx (10x/x) = (d/dx (10x) · x - 10x · d/dx(x)) / x²
We substitute d/dx (10x) = (10x) ln(10) and d/dx (x) = 1 = ((10x) ln(10) · x - 10x · 1) / x² = (x (10x) ln(10) - 10x) / x²
Therefore, d/dx (10x/x) = ((10x) (x ln(10) - 1)) / x²
Explanation
In this process, we differentiate the given function using the quotient rule. As a final step, we simplify the equation to obtain the final result.
FAQs on the Derivative of 10^x
1.Find the derivative of 10^x.
2.Can we use the derivative of 10^x in real life?
3.Is it possible to take the derivative of 10^x at any point?
4.What rule is used to differentiate 10^x/x?
5.Are the derivatives of 10^x and e^x the same?
Important Glossaries for the Derivative of 10^x
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Exponential Function: A function where a constant base is raised to a variable exponent.
- Natural Logarithm: The natural logarithm, denoted ln, is the logarithm to the base e.
- Quotient Rule: A rule used to differentiate functions that are divided by each other.
- Chain Rule: A rule used to differentiate composite functions.
Jaskaran Singh Saluja
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Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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