Problem 3 - Thomas Calculus

Question

In Exercises \(1-12,\) find \(d y / d x\) $$ y=\csc x-4 \sqrt{x}+7 $$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = -\csc x \cdot \cot x - 2x^{-1/2} \)

Step 1: Differentiate the first term

To find the derivative of the function \(y = \csc x - 4 \sqrt{x} + 7\), we start with the first term, \(\csc x\). The derivative of \(\csc x\) with respect to \(x\) is \(-\csc x \cdot \cot x\). Thus, \(\frac{d}{dx}(\csc x) = -\csc x \cdot \cot x\).

Step 2: Differentiate the second term

Next, differentiate the term \(-4\sqrt{x}\). This can be rewritten as \(-4x^{1/2}\). Use the power rule: \(\frac{d}{dx}x^n = nx^{n-1}\). Thus, the derivative is \(-4 \cdot \frac{1}{2}x^{-1/2} = -2x^{-1/2}\).

Step 3: Differentiate the third term

The third term is a constant, \(7\). The derivative of a constant is \(0\). Therefore, \(\frac{d}{dx}(7) = 0\).

Step 4: Combine the derivatives

Now, combine the derivatives of each term: \(\frac{dy}{dx} = -\csc x \cdot \cot x - 2x^{-1/2} + 0\). Simplifying, we get \(\frac{dy}{dx} = -\csc x \cdot \cot x - 2x^{-1/2}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in calculus, particularly when working with derivatives. Functions like sine (\( \sin \)), cosine (\( \cos \)), and cosecant (\( \csc \)) relate angles to ratios of sides in right-angled triangles. The function \( \csc x \) is the reciprocal of \( \sin x \). When differentiating trigonometric functions, it's crucial to remember their derivatives. The derivative of \( \csc x \) is \( -\csc x \cdot \cot x \). This result comes from combining the rules for differentiating reciprocal and composite functions.

• **Derivative of \( \csc x \)**: It results in \( -\csc x \cdot \cot x \), indicating both trigonometric functions' derivative outcomes.
• **Understanding cotangent**: \( \cot x \) is the reciprocal of tangent (\( \tan x \)) and equals \( \frac{1}{\tan x} = \frac{\cos x}{\sin x} \).

Understanding these derivatives helps to tackle more complex calculus problems involving trigonometric functions.

Power Rule

The power rule is a fundamental tool for finding derivatives of expressions with base variables raised to an exponent. It states that if you have a function \( x^n \), its derivative is \( nx^{n-1} \). This rule applies to any real number exponent.In this exercise, the term \( -4 \sqrt{x} \) is rewritten as \( -4x^{1/2} \) to make use of the power rule. By applying the power rule to \( x^{1/2} \), the derivative becomes \( \frac{1}{2}x^{-1/2} \). Multiplying by the original coefficient, \( -4 \), results in \( -2x^{-1/2} \).

• **Rewriting expressions**: Remember to convert radicals into exponents to apply the power rule easily.
• **Negative coefficients**: When differentiating, multiply the derivative by any constant coefficients outside the variable terms.

This method simplifies polynomial and rational expressions, making differentiation straightforward and systematic.

Derivative of Constants

The derivative of a constant, such as 7 in our exercise, is always zero. This holds because constants do not change, implying no change or slope across the graph. When solving derivatives, you can eliminate constant terms early, simplifying your work. Regardless of the complexity of the expression, this principle applies universally. Understanding this helps you focus on variable components that impact the derivative.

• **Simplicity principle**: Constants contribute zero to the rate of change, simplifying calculations.
• **Efficient calculus**: Managing constants efficiently can streamline finding derivatives.

Knowing this essential rule allows you to navigate derivatives smoothly, ensuring no unnecessary elements cloud your calculations.