[Solved] What is the Derivative of 10^x [10 to the x]?

The derivative of 10^x is equal to 10^x ln 10. Here ln denotes the logarithm with base e, called the natural logarithm. In this post, we will learn to compute the derivative of 10 raised to x.

Derivative of 10^x

Question: What is the derivative of 10^x?

Answer: The derivative of 10^x is 10^x ln 10.

Explanation:

To find the derivative of 10 to the x, we will use the logarithmic differentiation. Let us put

y=10^x

We need to find $\dfrac{dy}{dx}$. Taking logarithm on both sides with base 10, we get that

$\log_{10} y =\log_{10} 10^x$

⇒ $\log_{10} y =x$ as we know that log_a a^k=k.

Differentiating both sides with respect to x, we obtain that

$\dfrac{d}{dx}(\log_{10} y)=\dfrac{dx}{dx}$

⇒ $\dfrac{1}{\log_e 10} \dfrac{1}{y} \dfrac{dy}{dx}=1$

⇒ $\dfrac{dy}{dx}=y \cdot \log_e 10$

⇒ $\dfrac{dy}{dx}=10^x \cdot \ln 10$ as y=10^x and log_e=ln

Thus, the derivative of 10^x is 10^x ln 10.

Question-Answer on Derivative of 10^x

Question: What is the derivative of 10^x at x=0?

Answer:

From the above, we have that the derivative of 10^x is 10^x ln 10. That is,

$\dfrac{d}{dx}(10^x)=[10^x \ln 10]_{x=0}$

= 10⁰ ln 10

= 1 × ln 10 as we know that a⁰=1 for any non-zero number a.

= ln 10

So the derivative of 10 raised to x at x=0 is equal to ln10.

Also Read:

Derivative of 2^x

Derivative of 1/logx

Derivative of mod x

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