Question

What is the derivative of $ {5^{3x}} $ ?

Answer 1

Hint: In order to find the derivative of the given function, we should know which order derivative we need to find. Since, it’s not given so we would consider it as the first derivative. To find our first derivative of the function, we would use chain rule along with it we can see that the value is a constant with some power, so we would be using the formula: $ \dfrac{{d\left( {{a^x}} \right)}}{{dx}} = {a^x}\ln a $ and our derivative is obtained.

Complete step by step solution:
We are given with the function $ {5^{3x}} $ , which contains a constant that is $ 5 $ , with power of $ 3x $ .
From the formula of derivation for constants, we know that $ \dfrac{{d\left( {{a^x}} \right)}}{{dx}} = {a^x}\ln a $ , but in our function its not $ x $ , it’s $ 3x $ , which would be differentiated separately.
For these types of functions, we know that $ \dfrac{{d\left( {{a^u}} \right)}}{{dx}} = \left( {{a^u}\ln a} \right)\dfrac{{du}}{{dx}} $
Comparing $ u $ and $ a $ with our function $ {5^{3x}} $ , we get that $ u = 3x $ and $ a = 5 $ .
Substituting these values in the above formula, we get:
 $ \dfrac{{d\left( {{5^{3x}}} \right)}}{{dx}} = \left( {{5^{3x}}\ln 5} \right)\dfrac{{d\left( {3x} \right)}}{{dx}} $
On further solving with formulas of derivatives, we get:
 $
  \dfrac{{d\left( {{5^{3x}}} \right)}}{{dx}} = \left( {{5^{3x}}\ln 5} \right)\dfrac{{d\left( {3x} \right)}}{{dx}} \\
  \dfrac{{d\left( {{5^{3x}}} \right)}}{{dx}} = \left( {{5^{3x}}\ln 5} \right)\dfrac{{3d\left( x \right)}}{{dx}} \\
  \dfrac{{d\left( {{5^{3x}}} \right)}}{{dx}} = 3\left( {{5^{3x}}\ln 5} \right) \\
  \dfrac{{d\left( {{5^{3x}}} \right)}}{{dx}} = 3\left( {{5^{3x}}} \right)\ln 5 \\
  $
Therefore, our first derivative obtained is: $ \dfrac{{d\left( {{5^{3x}}} \right)}}{{dx}} = 3\left( {{5^{3x}}} \right)\ln 5 $
Hence, the derivative of $ {5^{3x}} $ is $ 3\left( {{5^{3x}}} \right)\ln 5 $ .
If it was given to find the second order derivative, we would have again derivated the function with respect to $ x $ , and similarly going on further for higher order.
So, the correct answer is “ $ 3\left( {{5^{3x}}} \right)\ln 5 $ ”.

Note: It’s always preferred to go step by step for derivating a function. If confident, can go for direct solving at a single step, but it creates a problem for larger functions in times of derivation.
It’s important to remember formulas of derivation to solve this type of question.
I. $ \dfrac{{d\left( {{a^x}} \right)}}{{dx}} = {a^x}\ln a $
II. $ \dfrac{{d\left( {{a^u}} \right)}}{{dx}} = \left( {{a^u}\ln a} \right)\dfrac{{du}}{{dx}} $
III. $ \dfrac{{d\left( {{x^n}} \right)}}{{dx}} = n{x^{n - 1}} $