Question

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=\frac{5}{(2 x)^{3}}+2 \cos x $$

Short answer

The derivative of the function \(y=\frac{5}{{(2x)}^{3}}+2\cos{x}\) is \(y'=-\frac{15}{(2x)^{4}}-2\sin{x}\).

Step-by-step solution

Identify the Given Function

We are given a function \(y = \frac{5}{{(2x)}^{3}} + 2 \cos{x}\), that is the sum of two different types of functions: a rational function and a trigonometric function. We need to find the derivative of this function.

Differentiate the First Part

Let's differentiate the first part of this function using the chain rule and the power rule. The derivative of \(y=\frac{5}{{(2x)}^{3}}\) with respect to \(x\) is \(-\frac{15}{(2 x)^{4}}\). Note here that \({(2x)}^n\) when differentiated equals to \(n \cdot 2 \cdot {({2x)}^{n-1}}\).

Differentiate the Second Part

Now we differentiate the second part of the function using the derivative of a cosine function. The derivative of \(2\cos{x}\) with respect to \(x\) is \(-2\sin{x}\). Note that the derivative of \(\cos{x}\) equals \(-\sin{x}\) and the coefficient in front simply comes along.

Combine the Derivatives

Finally, we combine together the derivatives of both parts of the function to get the derivative of the entire function, which is the sum of these two derivatives. Hence, \(y'=-\frac{15}{(2 x)^{4}}-2\sin{x}\).

Key concepts

Chain Rule

The chain rule is a fundamental technique in calculus used for finding the derivative of composite functions. A composite function is essentially a function nested within another function. The main idea is to differentiate the outer function, then multiply it by the derivative of the inner function. This process is crucial when tackling problems where a function depends on another function, especially when their forms are not easily separable.

For example, consider the function \( (2x)^3 \) in our exercise. Here, the outer function is \( u^3 \) and the inner function is \( u = 2x \). According to the chain rule, you would differentiate \( u^3 \) to get \( 3u^2 \) and then multiply it by the derivative of \( 2x \), which is 2. Applying the chain rule step-by-step ensures that all parts of the composite function are correctly differentiated.

Power Rule

The power rule simplifies finding the derivative of a function that is a power of \( x \). It states that if you have a term in the form of \( x^n \), its derivative is \( nx^{n-1} \). This rule is direct and fast, allowing ease when you have expressions raised to a power.

In our function, we applied the power rule to \( (2x)^3 \). First, treat \( 2x \) as a single entity to get \( 3(2x)^2 \). But since it’s not just \( x \), we also need to apply the chain rule to multiply by the derivative of \( 2x \), which is 2. Thus the full derivative using the power rule becomes \( 6(2x)^2 \).

Ultimately, this process reduces the complexity of differentiating powers especially when nested within other operations.

Trigonometric Differentiation

Trigonometric differentiation is used to find derivatives of trigonometric functions like sine, cosine, and tangent. Each of these functions has a straightforward derivative: the derivative of \( \cos{x} \) is \( -\sin{x} \), \( \sin{x} \) is \( \cos{x} \), and \( \tan{x} \) is \( \sec^2{x} \).

In our exercise, we differentiated the trigonometric function \( 2\cos{x} \). The derivative starts with \( \cos{x} \) becoming \( -\sin{x} \), and because there's a constant coefficient of 2 accompanying it, the result is \( -2\sin{x} \). Trigonometric differentiation simplifies the process by establishing these clear conversion rules between functions and their rates of change.

Rational Function Differentiation

Rational function differentiation involves finding the derivative of a function that can be expressed as a ratio of polynomials. This can often mean applying different rules of differentiation depending on how the function is structured. Generally, manipulating the function into a more differentiable form is a crucial first step.

Looking back at \( \frac{5}{(2x)^3} \), we rewrite it as \( 5(2x)^{-3} \) to facilitate easier differentiation. By shifting this format, we can apply the power rule more cleanly. The derivative of \( 5(2x)^{-3} \) is obtained by first using the power rule to get \( -15(2x)^{-4} \) and then applying the chain rule to account for \( 2x \). The end result reflects the derivatives of each component combined logically. This restructuring is essential in rational functions to handle complex numerators and denominators efficiently.