Problem 14 - Calculus for Biology and Medicine

Question

Find the derivative with respect to the independent variable. $$ f(x)=-3 \csc (3-5 x) $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 15 \csc(3-5x) \cot(3-5x) \).

Step 1: Identify the function and the rule to use

The given function is \( f(x) = -3 \csc(3 - 5x) \). To find the derivative, we will use the chain rule and the derivative of the cosecant function.

Step 2: Recall the derivative formula for the cosecant function

The derivative of \( \csc(u) \) with respect to \( u \) is \( -\csc(u) \cot(u) \). We will apply this formula to \( \csc(3 - 5x) \).

Step 3: Apply the chain rule

Using the chain rule, the derivative of \( -3 \csc(3 - 5x) \) is \[ f'(x) = -3 \left( -\csc(3-5x) \cot(3-5x) \right) \cdot \frac{d}{dx}(3-5x) \] where \( \frac{d}{dx}(3-5x) = -5 \).

Step 4: Simplify the expression

Compute the derivative:\[ f'(x) = -3 \cdot (-5) \csc(3-5x) \cot(3-5x) \]This simplifies to \[ f'(x) = 15 \csc(3-5x) \cot(3-5x) \]

Step 5: Conclude

The derivative of \( f(x) = -3 \csc(3-5x) \) is \[ f'(x) = 15 \csc(3-5x) \cot(3-5x) \]

Key Concepts

Chain Rule

The chain rule is a fundamental principle in calculus used to find the derivative of composite functions. A composite function is essentially a function within a function. If you can write a function as two nested functions, say \( f(g(x)) \), then it is composite. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function with respect to the variable. This is mathematically expressed as: \[ \frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x) \] To illustrate, consider \( f(x) = \sin(3x^2) \). Here, the outer function is \( \sin(u) \) where \( u = 3x^2 \). Using the chain rule:

• Find \( f'(u) = \cos(u) \)
• Find \( \frac{du}{dx} = 6x \)

Thus, the derivative \( \frac{d}{dx}\sin(3x^2) = \cos(3x^2) \cdot 6x \). By breaking a function into its composite parts, the chain rule provides us an effective method to calculate derivatives.

Trigonometric Functions

Trigonometric functions are mathematical functions related to angles of triangles and the unit circle. They are fundamental in calculus and have applications in many areas of science, including physics and engineering. The basic trigonometric functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), each of which relates the angles of a right triangle to ratios of its sides. Beyond the basics, we also have secant (\( \sec \)), cosecant (\( \csc \)), and cotangent (\( \cot \)), each being useful in calculus and higher mathematics. Importantly, these functions are periodic, meaning they repeat values in regular intervals. In calculus, these functions' derivatives are crucial. For example:

• The derivative of \( \sin(x) \) is \( \cos(x) \)
• The derivative of \( \cos(x) \) is \( -\sin(x) \)
• The derivative of \( \tan(x) \) is \( \sec^2(x) \)

Understanding these derivatives helps significantly when applying rules like the chain rule in trigonometric contexts.

Cosecant Function

The cosecant function is one of the lesser-known trigonometric functions but is important in solving calculus problems involving trig functions. Given by \( \csc(x) = \frac{1}{\sin(x)} \), it is the reciprocal of the sine function. This means it takes the inverse of the sine value; if sine of an angle is small, its cosecant is large, and vice versa. To differentiate \( \csc(x) \), we apply rules for reciprocals and trigonometric identities. The derivative of \( \csc(x) \) with respect to \( x \) is \[ \frac{d}{dx}\csc(x) = -\csc(x)\cot(x) \] Here, \( \cot(x) = \frac{\cos(x)}{\sin(x)} \), which simplifies calculations by using established derivatives. Knowing the derivatives of such reciprocal trig functions aids in handling more complex calculus problems that involve combinations of multiple trigonometric elements. The cosecant function, with its associated derivatives, allows for elegant solutions to intricate derivative problems.