Question

Differentiate the following functions. $$ u=\frac{\sin x}{a+b \cos x} $$

Short answer

Question: Find the derivative of the function \(u = \frac{\sin x}{a + b \cos x}\) with respect to \(x\). Answer: The derivative of the function \(u = \frac{\sin x}{a + b \cos x}\) with respect to \(x\) is given by: $$ \frac{du}{dx} = \frac{a\cos x + b}{(a+b \cos x)^2} $$

Step-by-step solution

Identify f(x) and g(x)

We have the function \(u = \frac{\sin x}{a + b \cos x}\). Here, \(f(x) = \sin x\) and \(g(x) = a + b \cos x\).

Find the derivatives f'(x) and g'(x)

To apply the Quotient Rule, we need to find the derivatives of \(f(x)\) and \(g(x)\). Let's calculate them: $$ f'(x) = \frac{d}{dx}(\sin x) = \cos x $$ $$ g'(x) = \frac{d}{dx}(a + b \cos x) = -b \sin x $$

Apply the Quotient Rule

Now that we have the derivatives of \(f(x)\) and \(g(x)\), let's apply the Quotient Rule to find the derivative of \(u\): $$ \frac{d}{dx}\left(\frac{\sin x}{a + b \cos x}\right)=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2} $$ Plugging in the derivatives and the functions, we get: $$ \frac{du}{dx} =\frac{(\cos x)(a+b \cos x)-(\sin x)(-b \sin x)}{(a+b \cos x)^2} $$

Simplify the expression

Now let's simplify the expression for the derivative: $$ \frac{du}{dx} = \frac{a\cos x+ b \cos^2 x + b \sin^2 x}{(a+b \cos x)^2} $$ We can use the trigonometric identity \(\sin^2 x + \cos^2 x = 1\) to further simplify the expression: $$ \frac{du}{dx} = \frac{a\cos x+ b (\sin^2 x + \cos^2 x)}{(a+b \cos x)^2} $$ Finally, our simplified derivative is: $$ \frac{du}{dx} = \frac{a\cos x + b}{(a+b \cos x)^2} $$

Key concepts

Quotient Rule

Differentiation of functions that are given as a quotient, where one function is divided by another, calls for a specific technique known as the Quotient Rule.
This rule is handy when you need to differentiate expressions like \( \frac{f(x)}{g(x)} \).

• First, identify the numerator as \( f(x) \) and the denominator as \( g(x) \).
• Next, find the derivatives \( f'(x) \) and \( g'(x) \).
• Finally, apply the formula: \[ \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \]

This formula ensures that you account for changes in both the top and bottom functions simultaneously, producing an accurate derivative of the quotient. It's crucial to perform each differentiation step methodically, ensuring both derivatives are acquired before substitution into the rule.

Trigonometric Derivatives

Trigonometric functions like \( \sin \), \( \cos \), and \( \tan \) have specific derivatives that are essential to know when dealing with calculus problems.

• The derivative of \( \sin x \) is \( \cos x \).
• For \( \cos x \), the derivative is \( -\sin x \).
• These derivatives arise naturally from the fundamental properties of sine and cosine functions.

Using these basic trigonometric derivatives helps in finding the derivatives for more complex expressions. When differentiating terms like \( a + b \cos x \), account for the constants in multiplication, applying the derivative directly to the trigonometric part, as done in finding \( g'(x) = -b \sin x \). Trigonometric derivatives are not just handy; they are necessary tools in the toolbox for solving a variety of differential calculus problems.

Simplification of Expressions

Derivatives often result in complex expressions. Simplifying them is key to making the result more comprehensible and practical for further use. To simplify, you look for common identities or terms that can be combined or reduced.

• One useful identity in trigonometry is \( \sin^2 x + \cos^2 x = 1 \). This allows for reduction of terms involving squared sines and cosines.
• In our function, replacing \( b \cos^2 x + b \sin^2 x \) with just \( b \) helps in achieving a cleaner form of the derivative.
• Simplifying also involves combining like terms and ensuring the expression is in its simplest form possible without complex fractions or unnecessary terms.

By simplifying, we turned the expression \( \frac{a \cos x + b \cos^2 x + b \sin^2 x}{(a + b \cos x)^2} \) into the more straightforward form \( \frac{a \cos x + b}{(a + b \cos x)^2} \). This step-wise simplification is integral to reducing errors and improving mathematical interpretation.