Derivative of 10^x
Answer
Derivative of 10^x
- Rule Used
- Exponential Rule
- Steps
- Recognize the exponential 10^x. For e^x the derivative equals the original function. For a^x the factor appears, because a^x = .
- Apply the rule and keep that constant factor.
- Final answer: 10^x*
d/dx 10^x = 10^x*ln(10). The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.
Instant step-by-step solution for 10^x
Derivative of 10^x = 10^x*ln(10) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-10-to-the-x
Derivative
10^x → 10^x*ln(10)engine: 10^x*ln(10)
Graph of the function and its derivative
The solid curve is f. The dashed curve is f′. Vertical gaps are asymptotes or jumps, not extra zeros.
Why the derivative has this form
10^x works like 2^x with a larger base. The extra factor is ln 10, about 2.302, so 10^x climbs faster than e^x. The derivative is 10^x ln 10, written 10^x*ln(10). The same rewrite, 10^x = e^(x ln 10), produces the factor; nothing about base 10 cancels it.
Apply the exponential rule to 10^x. The form to remember, already simplified, is 10^x*ln(10).
The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.
At x = 0, substitute into the simplified form 10^x*ln(10). That number is the slope of 10^x there; a secant over a small step in x lands next to it.
Worked derivative
- Write f(x) = 10^x.
- Apply the exponential rule to 10^x. The form to remember, already simplified, is 10^x*ln(10).
- At x = 0, substitute into the simplified form 10^x*ln(10). That number is the slope of 10^x there; a secant over a small step in x lands next to it.
Common mistakes
Answering 10^x instead of 10^x ln(10)
10^x alone would be right only for base e. Base 10 contributes ln 10, and that constant is not 1.
Treating 10^x as if only the base mattered
Only e^x is its own derivative with no extra factor. A coefficient in the exponent, or a base other than e, changes the answer.
Related rule and examples
- Exponential derivatives — The rule page lists this example and the others in the same family.
- Derivative of e^x — Same family of differentiation, different function.
- Derivative of e^(-x) — Same family of differentiation, different function.
- Derivative of e^(2x) — Same family of differentiation, different function.
- Derivative of 2^x — Same family of differentiation, different function.
Frequently asked questions
What is the derivative of 10^x?
10^x*ln(10). 10^x works like 2^x with a larger base. The extra factor is ln 10, about 2.302, so 10^x climbs faster than e^x. The derivative is 10^x ln 10, written 10^x*ln(10). The same rewrite, 10^x = e^(x ln 10), produces the factor; nothing about base 10 cancels it.
Which wrong answer shows up for 10^x?
10^x alone would be right only for base e. Base 10 contributes ln 10, and that constant is not 1.
Where is the derivative of 10^x defined?
The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.