Introduction to the derivative of 3^t

Derivatives have a wide range of applications in almost every field of engineering and science. The 3^t derivative is equal to the product of 3^t and log 3 which can be calculated by following the derivative rules. Or, we can directly find the derivative formula of 3^t by applying the first principle of differentiation. In this article, you will learn what the 3^t derivative formula is and how to calculate the derivatives of 3^t by using different approaches.

What is the derivative of 3^t?

The derivative of 3^t is equal to 3^t log 3. It measures the rate of change of the exponential function 3^t. It is denoted by d/dt(3^t) which is a fundamental concept in calculus.

Knowing the formula for derivatives and understanding how to use it can be used in solving problems related to velocity, acceleration, and optimization. 

Derivative of 3^t formula

The formula for derivative of f(x)=3^t is equal to the 3^t log 3, that is;

$f'(t) = \frac{d}{dt} (3^t) = 3^t log 3$

It is calculated by using the logarithmic differentiation.

How do you differentiate 3^t?

There are multiple derivative rules to prove the differentiation of 3 t. These are;

1. First Principle
2. Logarithmic differentiation
3. Quotient Rule

Each method provides a different way to compute the 3^t differentiation. By using these methods, we can mathematically prove the formula for finding the differential of 3^t.

Derivative of 3 power t by first principle

According to the first principle of derivative, the 3^t derivative is equal to 3^t log 3. The derivative of a function by first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,

$f’(x) = \lim_{h\to 0}\frac{f(x + h) - f(x)}{h}$

This formula allows us to determine the rate of change of a function at a specific point by using the limit definition of derivative.

Proof of 3^t derivative formula by first principle

To prove the derivative of 3^t by using first principle, replace f(x) by 3^t or you can replace it by 3^t to find the derivative of 3^n. 

$f’(t) = \lim_{h \to 0} \frac{f(t + h) - f(t)}{h}$

$f'(t) = \lim_{h \to 0} \frac{3^{t + h} - 3^t}{h}$

Moreover,

$f'(t) = \lim_{h \to 0} \frac{3^t+3^h-3^t}{h}$

$f'(t) = \lim_{h \to 0} \frac{3^t(3^h-1)}{h}$

When h approaches to zero,

$\lim_{h\to 0}\frac{3^h-1}{h}=\log 3$

And,

$f'(t) = 3^t \log 3$

Hence the differentiation of 3 t is equal to 3^t log 3. Use our derivative definition calculator with steps to simplify the above calculations easily. 

Derivative of 3^t by Logarithmic Differentiation

Logarithmic derivative is a technique of solving derivatives of logarithmic functions. A logarithmic function is the inverse of an exponential function and can be written using a base of 10. It is a method of finding derivatives of complex functions by applying logarithms.

Proof of differentiating of 3^t by Logarithmic Differentiation

To differentiate of 3^t by using the logarithmic differentiation, we start by assuming that,

$y = 3^t$

Taking log on both sides.

$\log y=\log 3^t$

$\log y=t \log 3$

Now differentiating on the both sides, 

$\frac{d}{dt} (\log y) = \log 3\frac{dt}{dt}$

Since the derivative of log x is equal to the reciprocal of x, therefore, 

$\frac{dy}{dt}.\frac{1}{y}= \log 3$

Hence,

$\frac{dy}{dt} = y\log 3$

Substituting the value of y, we get

$\frac{dy}{dt} = 3^t\log 3$

Hence we can derive the derivative of 3 t by using two derivative rules i.e. delta method and the logarithmic differentiation. 

How to find the derivatives of 3^t with a calculator?

The easiest way of differentiating 3^t is by using an online calculator. You can use our derivative calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

1. Write the function as 3^t in the enter function box. In this step, you need to provide input value as a function that you want to differentiate.
2. Now, select the variable by which you want to differentiate 3^t. Here you have to choose n.
3. Select how many times you want to calculate the derivatives of 3^t. In this step, you can choose 2 to calculate the second derivative, 3 for the third derivative and so on.
4. Click on the calculate button. After this step, you will get the derivative of 3 to the t within a few seconds.

After completing these steps, you will receive the differential of 3 t within seconds. Using online tools like dy/dx calculator can make it much easier and faster to calculate derivatives, especially for complex functions.

Conclusion:

In conclusion, the derivative of 3^t is 3^t log 3. The derivative measures the rate of change of a function with respect to its independent variable, and in the case of 3^t, the derivative can be calculated using the first principle of derivative and the logarithmic differentiation.