Скачать или смотреть The EASIEST Way to Find Derivatives with Natural Logs!

Hello! In this problem we are asked to find the derivative of the function Logarithm of x to the power tangent of x .
We will carefully walk you through the solution, by applying the chain rule, and the logarithmic differentiation technique.
In this problem, we have to compute the derivative of Logarithm of x to the power tangent of x .
notice that ,, This function can be seen as an exponential function because it is obtained by raising Logarithm of x to the power tangent of x .
In this case Logarithm of x is seen as the base and tangent of x is the exponent. We will have to use the fact that the exponential and the logarithmic functions are inverse functions.
Then apply the chain rule to find the derivative.
In order to simplify this expression, we will use the properties of the exponential and logarithmic function. We know that, The exponential of log of x is equal x which is also equal to the logarithm of the exponential of x.
Please do not forget to like the video, and subscribe to our Channel. It will help the growth of our channel.
see that , we will also have to use the chain rule. Let us recall the chain rule.
The derivative of the composition of two functions u of v of x is equal to v prime of x times, u prime of v of x.
Then, , we express f of x as exponential of logarithm of Logarithm of x to the power tangent of x .
Using the properties of the exponential and the logarithm, actually f of x equals exponential of tangent of x times, logarithm of Logarithm of x . In fact, we are using the fact that the logarithmic of x to the power y is equal to y times, the logarithm of x.
Now, we can take the derivative of both sides and use the chain rule.
actually , using the chain rule, the derivative of f is equal to the derivative of the product tangent of x , times, logarithm of Logarithm of x , times exponential of the product tangent of x , times the logarithm of Logarithm of x
Actually , in order to compute the derivative of the product tangent of x , times logarithm of Logarithm of x , we need to use the product rule. Let us recall the product rule.
The derivative of the product of two functions u of x times v of x is equal to u prime of x times v of x plus,
u of x times v prime of x
see that , in order to find the derivative of the quantity tangent of x times, logarithm of Logarithm of x , we need to use the product rule. In this case, the derivative is tangent of x , times, the fraction 1 over x over Logarithm of x , + secant square of x , times logarithm of Logarithm of x
ultimately, , f prime of x is equal to the quantity tangent of x , times, the fraction 1 over x over Logarithm of x , + secant square of x , times logarithm of Logarithm of x , times the exponential of tangent of x , times the logarithm of Logarithm of x .
This final expression can also be given as follows; f prime of x is equal to the quantity tangent of x , times, the fraction 1 over x over Logarithm of x , + secant square of x , times logarithm of Logarithm of x , times Logarithm of x to the power tangent of x .
Thank you for viewing this video. Please subscribe to our Channel. We have an extensive collection of calculus-related videos covering limits, derivatives, integration and many other calculus topics