Question

Find the derivative of the transcendental function. $$ f(\theta)=(\theta+1) \cos \theta $$

Short answer

The derivative of the function \(f(\theta)=(\theta+1) \cos \theta\) is \(f'(\theta) = \cos(\theta) - (\theta +1) \sin \theta\).

Step-by-step solution

Identify the functions

We first identify the two different functions which are being multiplied together. Here they are \(\theta+1\) (let's call this function \(u\)) and \(\cos \theta\) (let's call this function \(v\)).

Apply the Product Rule

The Product Rule states that the derivative of two multiplied functions is the first function times the derivative of the second function plus the second function times the derivative of the first function. In this case, the derivative can be written as: \(f'(\theta) = u'(\theta)v(\theta) + u(\theta)v'(\theta)\), where \(u'(\theta)\) is the derivative of \(\theta+1\) and \(v'(\theta)\) is the derivative of \(\cos \theta\).

Calculate the derivatives

Now, compute the derivatives of \(u\) and \(v\). The derivative \(u'(\theta)\) of \(\theta+1\) is 1 and the derivative \(v'(\theta)\) of \(\cos \theta\) is \(-\sin \theta\).

Substitute the derivatives into the Product Rule equation

Now substitute the values of \(u(\theta)\), \(v(\theta)\), \(u'(\theta)\) and \(v'(\theta)\) in our equation from Step 2. This gives us: \(f'(\theta) = 1\cdot \cos(\theta) + (\theta +1)(-\sin \theta)\).

Key concepts

Product Rule

The Product Rule is a fundamental concept in calculus for finding the derivative of a product of two functions. It's essential when you're dealing with equations where one variable is the product of two or more others. In simple terms, if you have a function that can be expressed as the multiplication of two sub-functions, like \( f(x) = u(x) \times v(x) \), the derivative of \( f \) with respect to \( x \) is found using the Product Rule.

The Product Rule tells us that \( f'(x) = u'(x)v(x) + u(x)v'(x) \). Here, \( u'(x) \) and \( v'(x) \) are the derivatives of \( u \) and \( v \) respectively. Instead of trying to simplify the product first, you differentiate each function independently and then apply the Product Rule. This rule is particularly useful because it simplifies the process of differentiation, and it's applicable to a wide range of problems.

Differentiation

Differentiation is the process of finding the derivative of a function, which is the rate at which the function's value changes. When you differentiate a function, you are essentially looking for the slope of the function at any given point. Differentiation can involve simple functions, where you apply basic rules, or more complex functions that require advanced techniques, such as the Product Rule or Chain Rule.

For a basic function like \( f(x)=x^n \), the power rule states that the derivative is \( f'(x)=nx^{n-1} \). However, for products or composite functions, you cannot apply the power rule directly. It's important to remember that rules of differentiation are tools to help solve these kinds of problems. With practice, identifying which rule to apply becomes more intuitive.

Transcendental Functions

Transcendental functions, such as exponential, logarithmic, trigonometric, and inverse trigonometric functions, go beyond algebraic functions because they cannot be expressed as a finite sequence of the algebraic operations of addition, multiplication, and root extraction.

Common Transcendental Functions

• Exponential functions like \( e^x \) or \( a^x \) where \( a \) is a constant.
• Logarithmic functions like \( \text{ln}(x) \) or \( \text{log}_a(x) \) where \( a \) is the base of the logarithm.
• Trigonometric functions such as \( \text{sin}(x) \), \( \text{cos}(x) \), and \( \text{tan}(x) \).
• Inverse Trigonometric functions like \( \text{arcsin}(x) \), \( \text{arccos}(x) \), and \( \text{arctan}(x) \).

These functions often model more complex phenomena in physics, engineering, and other fields, which makes understanding their differentiation critical.

Chain Rule

The Chain Rule is another essential tool in calculus used for finding the derivative of composite functions. If you have a function \( h(x) \) that is the composition of two functions \( f \) and \( g \), such that \( h(x) = f(g(x)) \), the derivative of \( h \) is not just the product of the derivatives of \( f \) and \( g \). Instead, you use the Chain Rule, which lets you find the derivative of \( h \) by taking the derivative of \( f \) with respect to \( g \), and then multiplying it by the derivative of \( g \) with respect to \( x \).

In formulaic terms, if \( h(x) = f(g(x)) \), then the derivative \( h'(x) \) is \( f'(g(x)) \times g'(x) \). This rule allows you to differentiate a wide range of composite functions where one function is nested inside another.