Question

Evaluate the following derivatives. $$\frac{d}{d x}\left(x^{\tan x}\right)$$

Short answer

Question: Find the derivative of the function \(f(x) = x^{\tan x}\) with respect to \(x\). Answer: \(\frac{dy}{dx} = x^{\tan x} \left( \sec^2 x \ln x + \tan x \frac{1}{x}\right)\)

Step-by-step solution

Rewrite function in terms of y

Let \(y = x^{\tan x}\). We are going to find the derivative of \(y\) with respect to \(x\).

Take the natural logarithm of both sides

Taking the natural logarithm of both sides of the equation, we get $$\ln y = \ln (x^{\tan x}).$$

Apply logarithm properties

Using the logarithm property \(\ln(a^b) = b\ln a\), we can rewrite the equation as $$\ln y = \tan x \ln x.$$

Differentiate implicitly with respect to x

Now, we differentiate both sides of the equation with respect to \(x\). We will need to use the product rule on the right side and the chain rule on the left side: $$\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} (\tan x \ln x)$$

Apply the product rule

Differentiating the right side using the product rule, we get $$\frac{1}{y} \frac{dy}{dx} = \left(\frac{d}{dx} \tan x \right)\ln x + \tan x \left(\frac{d}{dx}\ln x \right)$$

Compute the derivatives

We know that the derivative of \(\tan x\) is \(\sec^2 x\). The derivative of \(\ln x\) is \(\frac{1}{x}\). So we can write the equation as $$\frac{1}{y} \frac{dy}{dx} = \sec^2 x \ln x + \tan x \frac{1}{x}$$

Isolate the desired derivative

Now, we can solve for the desired derivative, \(\frac{dy}{dx}\), by multiplying both sides of the equation by \(y\): $$\frac{dy}{dx} = y \left(\sec^2 x \ln x + \tan x \frac{1}{x}\right)$$

Substitute y back in terms of x

Recall that \(y = x^{\tan x}\). Substituting this back into the equation, we get $$\frac{dy}{dx} = x^{\tan x} \left( \sec^2 x \ln x + \tan x \frac{1}{x}\right)$$ This is the final expression for the derivative of the function \(f(x) = x^{\tan x}\) with respect to \(x\).

Key concepts

Implicit Differentiation

Implicit differentiation is a technique used when dealing with functions that are not explicitly defined. Here, instead of directly expressing one variable as a function of another, both variables are intermingled in an equation. This requires a different approach for finding derivatives.

In the exercise, we started with the equation \( y = x^{\tan x} \). To find its derivative, we had to involve both variables \( y \) and \( x \) as they are not given in an explicit relation. This led us to first express the problem in terms of \( y \), and then differentiate both sides concerning \( x \).

By differentiating implicitly, we take the derivative of the entire equation, keeping in mind to apply the chain rule or other necessary differentiation techniques for each part of the equation.

Product Rule

The product rule is a must-know tool for differentiation. This rule is essential when you have to differentiate expressions where two or more functions are multiplied together.

If you have to differentiate \( u(x) \cdot v(x) \), where \( u \) and \( v \) are both functions of \( x \), the derivative is given by:

• \( \frac{d}{dx}[u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) \)

In our exercise with \( \tan x \ln x \), this rule was employed. We treated \( \tan x \) as one function and \( \ln x \) as another, resulting in the decomposition of derivatives per the product rule. This essential step allowed us to find the derivative of the entire expression efficiently.

Chain Rule

The chain rule is a fundamental principle that applies when you have functions within functions. It is used to differentiate composite functions.

• Think of it like peeling layers of an onion, differentiating each layer one at a time.
• If you have a function \( f \) made up of a function \( g(x) \) inside another function \( h(g(x)) \), then the derivative is \( f'(x) = h'(g(x)) \cdot g'(x) \).

In our context, the chain rule was used when differentiating \( \ln y \) concerning \( x \). Since \( y \) is a function of \( x \), we utilized the chain rule to ensure that each component part of \( \ln y \)'s derivative was accounted for, including the hidden dependency on \( x \).

Logarithm Properties

Logarithm properties are very useful when dealing with exponentiation and derivatives. One of the crucial properties applied in this exercise was \( \ln(a^b) = b \ln a \), which helped simplify expressions.

By rewriting \( y = x^{\tan x} \) as \( \ln y = \tan x \ln x \), we reduced the complexity of differentiation. Working with a product of simple expressions like \( \tan x \ln x \) is often much more straightforward than tackling the original exponential format directly.

Using logarithm properties, we can transform complex, non-linear expressions into more manageable linear ones, simplifying the differentiation journey considerably.