[Solved] What is the Derivative of 5^x [5 to the power x]?

The derivative of 5^x is equal to 5^x ln5. Here, ln denotes the natural logarithm (logarithm with base e). In this post, we will find the derivative of 5 to the power x.

Derivative of 5^x Formula

As we know that the derivative of a^x is a^x ln a, the formula for the derivative of 5^x will be as follows:

$\dfrac{d}{dx}(5^x)=5^x \ln 5$

or

$(5^x)’=5^x \ln 5$.

Here, the prime $’$ denotes the first-order derivative.

What is the Derivative of 5^x?

Answer: The derivative of 5^x is 5^xln5.

Explanation:

Let us find the derivative of 5 raised to x by the logarithmic differentiation. To do so, we put

$z=5^x$.

Taking natural logarithms $\ln$ of both sides, we obtain that

$\ln z=\ln 5^x$

$\Rightarrow \ln z=x\ln 5$

Differentiating both sides with respect to x, we get that

$\dfrac{1}{z} \dfrac{dz}{dx}=\ln 5$

$\Rightarrow \dfrac{dz}{dx}=z\ln 5$

$\Rightarrow \dfrac{dz}{dx}=5^x\ln 5$ as $z=5^x$.

Thus, the derivative of 5^x is 5^x ln5.

Also Read: 

Derivative of e²

Derivative of Pi

Derivative of log(3x)

Derivative of sin 4x

Derivative of tan 3x

Derivative of 5^x at x=1

From the above, we get that the derivative of $5^x$ is equal to $5^x \ln 5$. So the derivative of 5 to the power x at x=1 will be equal to

$\dfrac{d}{dx}[5^x]{x=1}$

$=[5^x \ln 5]{x=1}$

$=5^1 \ln 5$

$=5 \ln 5$.

Thus, the derivative of 5^x at x=1 is 5ln 5.

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Derivative of sin⁵ x

Derivative of cos⁴ x

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