Problem 36 - Single Variable Calculus

Question

Find the derivative of the function. $$ z=\left(1+\csc ^{2} x\right)^{4} $$

Step-by-Step Solution

Answer

The short answer is: \(\frac{dz}{dx} = - 8(1 + \csc^2{x})^3 \cdot \csc^2{x}\cot{x}\).

Step 1: Identify the inner and outer functions

In this exercise, the given function is \(z = (1 + \csc^2{x})^4\). We can identify the inner function \(u\) as \(1 + \csc^2{x}\) and the outer function as \(z = u^4\).

Step 2: Find the derivative of the outer function

The outer function is a simple power function \(z = u^4\). To find its derivative with respect to \(u\), we apply the power rule, which states that the derivative of a function \(z = u^n\) is \(\frac{dz}{du} = nu^{n-1}\). In this case, with n = 4: \(\frac{dz}{du} = 4u^3\)

Step 3: Find the derivative of the inner function

The inner function is \(u = 1 + \csc^2{x}\) and its derivative with respect to \(x\) can be found by applying the derivative of the cosecant function. The derivative of the cosecant function is given by \(\frac{d}{dx}(\csc{x}) = -\csc{x}\cot{x}\). So, we have the derivative of the inner function as follows: \(\frac{du}{dx} = 0 - 2\csc{x}\cot{x}\csc{x} = - 2\csc^2{x}\cot{x}\)

Step 4: Apply the chain rule to find the derivative of the given function

Now that we have found the derivatives of the inner and outer functions, we have all the components needed to apply the chain rule: \(\frac{dz}{dx} = \frac{dz}{du} \cdot \frac{du}{dx} = 4u^3 \cdot (- 2\csc^2{x}\cot{x})\) However, recall that the inner function is \(u = 1 + \csc^2{x}\), so we can substitute it back into the expression above to get the final result: \(\frac{dz}{dx} = - 8(1 + \csc^2{x})^3 \cdot \csc^2{x}\cot{x}\)

Step 5: Simplify the result

Our final answer, the derivative of the original function with respect to \(x\), is: \(\frac{dz}{dx} = - 8(1 + \csc^2{x})^3 \cdot \csc^2{x}\cot{x}\)

Key Concepts

Derivative of Power Function

When dealing with calculus, one of the essential rules students must learn is the power rule for differentiation. This rule applies to functions of the form \( f(x) = x^n \), where \( n \) is any real number. The power rule states that the derivative of \( f(x) \) with respect to \( x \) is \( f'(x) = nx^{n-1} \).

For example, to differentiate a function like \( u^4 \), simply bring down the exponent as a coefficient and decrease the exponent by one, which results in the derivative \( 4u^3 \). This technique is particularly useful because it requires minimal calculation and applies to all power functions, making it a fundamental concept in calculus.

Derivative of Cosecant Function

The cosecant function, denoted as \( \text{csc}(x) \), is the reciprocal of the sine function. Differentiation of trigonometric functions is standard in calculus, and the cosecant function has its unique rule. The derivative of \( \text{csc}(x) \) is given by \( -\text{csc}(x)\text{cot}(x) \), where \( \text{cot}(x) \) stands for the cotangent function, the reciprocal of the tangent function.

Understanding this derivative is crucial when the cosecant function appears in more complex expressions. It is important to remember the negative sign, which is often overlooked but critical in reaching the correct result.

Chain Rule in Calculus

The chain rule is a powerful differentiation technique used when dealing with compositions of functions. Essentially, the chain rule states that to find the derivative of a composite function \( f(g(x)) \), one should take the derivative of the outer function \( f(u) \), where \( u = g(x) \), and then multiply it by the derivative of the inner function \( g(x) \). Mathematically, it is expressed as \( \frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} \).

In our case, the composite function is \( (1 + \text{csc}^2(x))^4 \). To differentiate it, we first treat \( 1 + \text{csc}^2(x) \) as the inner function and find its derivative. Then, we multiply it by the derivative of the outer function \( u^4 \) to obtain the final derivative of the composite function. By mastering the chain rule, students can efficiently handle complex differentiation problems.

Differentiation Techniques

Differentiation techniques consist of numerous rules and methods to determine the derivative of a function. These include fundamental rules such as the product rule, quotient rule, and chain rule, as well as techniques for differentiating trigonometric, exponential, logarithmic, and inverse functions.

To solve diverse calculus problems effectively, students should be familiar with these techniques and know when to apply each one. For trigonometric functions like the cosecant, specific rules apply, while power functions often call for the power rule. A thorough grasp of these techniques ensures students can tackle any differentiation challenge they encounter.