Question

Verify the following derivative formulas using the Quotient Rule. $$\frac{d}{d x}(\csc x)=-\csc x \cot x$$

Short answer

Question: Verify the derivative formula: $\frac{d}{d x}(\csc x) = -\csc x \cot x$ Solution: Step 1: Given that $\csc x = \frac{1}{\sin x}$, we found the derivatives of $g(x) = 1$ and $h(x) = \sin x$. We obtained $g'(x) = 0$ and $h'(x) = \cos x$. Step 2: Using the Quotient Rule, we plugged the values of $g(x)$, $h(x)$, $g'(x)$, and $h'(x)$ into the formula, getting: $\frac{d}{d x}(\csc x) = \frac{-\cos x}{\sin^2 x}$. Step 3: Simplifying the expression and using the definitions of $\csc x$ and $\cot x$, we got: $\frac{d}{d x}(\csc x) = -\csc x \cot x$. Step 4: We verified that the derived formula matches the given formula, proving its correctness.

Step-by-step solution

Find the derivatives of g(x) and h(x)

We need to find the derivatives of \(g(x) = 1\) and \(h(x) = \sin x\). For \(g(x) = 1\): Since the function is a constant, its derivative is zero: $$g'(x) = \frac{d}{d x}(1) = 0$$ For \(h(x) = \sin x\): We know the derivative of \(\sin x\) is: $$h'(x) = \frac{d}{d x}(\sin x) = \cos x$$

Apply the Quotient Rule

Now, we'll plug in our functions and their derivatives into the Quotient Rule formula: $$\frac{d}{d x}(\csc x) = \frac{d}{d x}\left(\frac{1}{\sin x}\right) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}$$ Substitute the values of \(g(x)\), \(h(x)\), \(g'(x)\), and \(h'(x)\) from Step 1: $$\frac{d}{d x}(\csc x) = \frac{0 \cdot \sin x - 1 \cdot \cos x}{[\sin x]^2}$$

Simplify the expression

Simplify the expression to match the given formula: $$\frac{d}{d x}(\csc x) = \frac{-\cos x}{\sin^2 x}$$ We can rewrite this expression using the definitions of \(\csc x\) and \(\cot x\). Recall that \(\csc x = \frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\): $$\frac{d}{d x}(\csc x) = -\csc x \cot x$$

Verify the formula

We have derived the derivative of \(\csc x\) using the Quotient Rule and found that: $$\frac{d}{d x}(\csc x) = -\csc x \cot x$$ This matches the given formula, verifying its correctness.

Key concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It's divided into two main parts: differential calculus, which concerns itself with derivatives and rates of change, and integral calculus, which focuses on accumulation and areas under curves.

When solving problems involving derivatives—such as finding the rate of change of a cosecant function—it's crucial to understand not just the procedures but also the underlying concepts. For instance, why do we care about rates of change? In the physical world, calculus helps us predict and understand phenomena like velocity and acceleration, growth rates, and more. By learning calculus, students develop a toolkit for modeling and solving a vast array of mathematical and real-world problems.

Derivative of Cosecant

The derivative of the cosecant function is not immediately obvious, but with some knowledge of trigonometry and calculus, it can be derived using fundamental rules.

The cosecant function, written as \(\csc x\), is the reciprocal of the sine function (\

Simplifying Expressions

Simplifying mathematical expressions is an indispensable skill in calculus, making expressions easier to understand and work with. The simplification process often involves algebraic manipulations such as combining like terms, factoring, and using trigonometric identities.

For example, simplifying the expression \(\frac{-\cos x}{\sin^2 x}\) into \( -\csc x \cot x\) requires the recognition of trigonometric identities. Understanding this process allows for more efficient problem-solving and can reveal insights into the behavior of mathematical functions. Simplification isn't just a mechanical process; it's a critical analytical tool in calculus.

Applying Derivative Rules

Applying derivative rules effectively is a key component of mastering calculus. Derivatives can be found using a range of rules, such as the Power Rule, Product Rule, Chain Rule, and the Quotient Rule. To determine the derivative of complex expressions, it's necessary to decide which rule or combination of rules to apply, and in what order.

The Quotient Rule, used for differentiating a function that is the quotient of two other functions, states that the derivative of \(\frac{g(x)}{h(x)}\) is \(\frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}\), where \(g'(x)\) and \(h'(x)\) are the derivatives of \(g(x)\) and \(h(x)\), respectively. Proper application of these rules, combined with simplification of the result, reveals derivatives that form the foundation for understanding the change and motion described by calculus.