Question

Differentiate the following functions. $$ u=\frac{1}{a \cos x+b \sin x} $$

Short answer

Based on the solution provided above, the derivative of the function \(u = \frac{1}{a\cos{x} + b\sin{x}}\) with respect to x is given by: $$ \frac{du}{dx} = \frac{a\sin{x} - b\cos{x}}{(a\cos{x} + b\sin{x})^2} $$

Step-by-step solution

Identify the top and bottom functions

Let's identify the top function (numerator) and the bottom function (denominator) from the given function. In this case, the top function is \(1\) and the bottom function is \((a\cos{x}+b\sin{x})\).

Apply quotient rule for derivatives

The quotient rule formula for finding the derivative of \(\frac{f(x)}{g(x)}\) is as follows: $$ \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} $$ We will use this formula for finding the derivative of our function. In our case, f(x) = 1 and g(x) = \((a\cos{x}+b\sin{x})\).

Compute derivatives of f(x) and g(x)

In order to apply the quotient rule, we need to find the derivatives of f(x) and g(x) with respect to x. Since \(f(x) = 1\), the derivative of f(x) with respect to x is: $$f'(x) = \frac{d(1)}{dx} = 0$$ On the other hand, g(x) is a combination of trigonometric functions, so we'll differentiate each of them separately using chain rule: $$ g'(x) = \frac{d(a\cos{x}+b\sin{x})}{dx} = -a\sin{x} + b\cos{x} $$

Apply the quotient rule formula

Now we have everything we need to apply the quotient rule formula. Plugging in the values of f(x), f'(x), g(x), and g'(x), we get: $$ \frac{du}{dx} = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} = \frac{0\cdot(a\cos{x} + b\sin{x}) - (1)\cdot(-a\sin{x} + b\cos{x})}{(a\cos{x} + b\sin{x})^2} $$

Simplify the result

Simplify the obtained expression: $$ \frac{du}{dx} = \frac{a\sin{x} - b\cos{x}}{(a\cos{x} + b\sin{x})^2} $$ Now we have found the derivative of the given function with respect to x.

Key concepts

Quotient Rule

Understanding how to differentiate a function that consists of one function divided by another is crucial in calculus, and this is where the quotient rule comes into play. The quotient rule is a formula used to find the derivative of a quotient of two functions. It states that if you have a function in the form of \(u = \frac{f(x)}{g(x)}\), the derivative \(u'\) is calculated as:
\[\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}\]
This can seem daunting at first, but breaking it down helps. The numerator of the resulting fraction involves finding the derivative of the top function (\(f'(x)\)) and multiplying it by the bottom function (\(g(x)\)) untouched, then subtracting the top function (\(f(x)\)) multiplied by the derivative of the bottom (\(g'(x)\)). The denominator is simply the bottom function squared. Remember to simplify the derivative expression where possible for a clearer result.

Derivatives of Trigonometric Functions

Differentiating trigonometric functions is a frequent task in calculus. It's vital to remember the basic derivatives of sine and cosine functions: \(\frac{d}{dx}(\sin x) = \cos x\) and \(\frac{d}{dx}(\cos x) = -\sin x\).
In the provided exercise, the derivative of a linear combination of a cosine and a sine function is required. This derivative is found by differentiating each term separately: \(g'(x) = \frac{d(a\cos{x}+b\sin{x})}{dx} = -a\sin{x} + b\cos{x}\). These trigonometric derivatives are foundational and serve as building blocks for more complex problems involving trigonometric functions. Always apply these rules correctly to get accurate results when differentiating.

Simplifying Derivatives

Once we have applied the rules for differentiation, such as the quotient rule, we often end up with complex expressions. Simplifying these derivatives makes them easier to understand and further work with. In our exercise, after using the quotient rule, we reached the derivative
\[\frac{du}{dx} = \frac{a\sin{x} - b\cos{x}}{(a\cos{x} + b\sin{x})^2}\]
Simplification may include combining like terms, factoring, canceling common factors, and reducing fractions. Simplified derivatives are not only easier to comprehend but also necessary for computing accurate values of derivatives, plotting graphs of derivatives, or solving equations involving derivatives. Try simplifying step by step, checking for common factors or identities that could further reduce the result.

Applying Chain Rule

The chain rule is a powerful tool for computing the derivative of the composition of two or more functions. It states that if you have a composite function \(h(x) = f(g(x))\), its derivative is the derivative of \(f\) with respect to \(g\) multiplied by the derivative of \(g\) with respect to \(x\):
\[h'(x) = f'(g(x)) \cdot g'(x)\]
When differentiating trigonometric functions that are multiplied by a constant, as seen in our exercise with \(a\cos{x}\) and \(b\sin{x}\), the constants simply carry through the differentiation process, like coefficients. This is an application of the chain rule, where the outside function is the trigonometric function, and the inside function is the linear function \(x\). Applying the chain rule correctly is critical for the correct differentiation of composite functions.