Problem 35 - Calculus for Biology and Medicine

Question

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\frac{\sec \left(x^{2}-1\right)}{\csc \left(x^{2}+1\right)} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{2x \cdot \cos(2)}{(\cos(x^2 - 1))^2} \).

Step 1: Rewrite the Function with Trigonometric Identities

We start by rewriting the function in terms of sine and cosine to simplify the differentiation process. Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} \) and \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Thus, the function can be expressed as:\[ f(x) = \frac{\frac{1}{\cos(x^2 - 1)}}{\frac{1}{\sin(x^2 + 1)}} = \frac{\sin(x^2 + 1)}{\cos(x^2 - 1)} \]

Step 2: Apply the Quotient Rule

The function can now be differentiated using the quotient rule. The quotient rule is given by:\[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \]For our function, let \( u = \sin(x^2 + 1) \) and \( v = \cos(x^2 - 1) \).

Step 3: Differentiate the Numerator and Denominator

First, find the derivative \( \frac{du}{dx} \). Using the chain rule, this is:\[ \frac{d}{dx} \sin(x^2 + 1) = \cos(x^2 + 1) \cdot \frac{d}{dx}(x^2 + 1) = \cos(x^2 + 1) \cdot 2x \]Next, find \( \frac{dv}{dx} \):\[ \frac{d}{dx} \cos(x^2 - 1) = -\sin(x^2 - 1) \cdot \frac{d}{dx}(x^2 - 1) = -\sin(x^2 - 1) \cdot 2x \]

Step 4: Substitute into the Quotient Rule

Now substitute \( \frac{du}{dx} \) and \( \frac{dv}{dx} \) back into the quotient rule:\[ \frac{d}{dx} \frac{\sin(x^2 + 1)}{\cos(x^2 - 1)} = \frac{\cos(x^2 - 1) \cdot (\cos(x^2 + 1) \cdot 2x) - \sin(x^2 + 1) \cdot (-\sin(x^2 - 1) \cdot 2x)}{(\cos(x^2 - 1))^2} \]

Step 5: Simplify the Expression

Simplifying the terms in the numerator, we get:\[ 2x \left( \cos(x^2 + 1) \cos(x^2 - 1) + \sin(x^2 + 1) \sin(x^2 - 1) \right) \]Now, notice that this can be simplified using the angle addition formula:\( \cos(A - B) = \cos A \cos B + \sin A \sin B \). Therefore, \[ 2x \cdot \cos((x^2 + 1) - (x^2 - 1)) = 2x \cdot \cos(2) \]The derivative simplifies to:\[ \frac{2x \cdot \cos(2)}{(\cos(x^2 - 1))^2} \]

Key Concepts

Trigonometric Identities

In calculus, simplifying expressions using trigonometric identities can make differentiation much more straightforward. Trigonometric identities are essential tools that allow us to transform complex trigonometric expressions into simpler forms. For instance, the secant and cosecant functions can be rewritten using the basic sine and cosine functions.

• Secant ( c) is defined as the reciprocal of cosine: \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
• Cosecant is similarly the reciprocal of sine: \( \csc(\theta) = \frac{1}{\sin(\theta)} \).

In the given function \( f(x) = \frac{\sec(x^2 - 1)}{\csc(x^2 + 1)} \), these identities transform it into a more manageable form for differentiation. The function simplifies to \( \frac{\sin(x^2 + 1)}{\cos(x^2 - 1)} \), using the principle that dividing by a fraction is the same as multiplying by its reciprocal. This highlights how trigonometric identities can drastically reduce the complexity of functions we need to differentiate.

Quotient Rule

In calculus, when dealing with functions expressed as a quotient of two differentiable functions, the quotient rule is employed to find their derivatives. It is given by:\[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \]This rule allows us to take the derivative of a quotient by separately differentiating the numerator and the denominator. For the function \( f(x) = \frac{\sin(x^2 + 1)}{\cos(x^2 - 1)} \), we apply the quotient rule by letting:

• \( u = \sin(x^2 + 1) \)
• \( v = \cos(x^2 - 1) \)

Both components are differentiated individually, and then substituted back into the formula. This systematic approach helps prevent common differentiation errors, ensuring that every component of the function is processed correctly. Using the quotient rule, we differentiate the expression step-by-step without losing sight of each part's role in the overall function.

Chain Rule

The chain rule is a fundamental derivative rule employed in calculus to differentiate composite functions. When a function is nested within another, the chain rule is the strategy we use. Its formula is:\[ \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \]This principle allows us to differentiate indirectly by multiplying the derivative of the outer function by the derivative of the inner function. For example, in the expression \( \sin(x^2 + 1) \), the outer function is \( \sin(u) \) and the inner function is \( u = x^2 + 1 \).

• Differentiating the outer function \( \sin(u) \) yields \( \cos(u) \).
• Differentiating the inner function \( u = x^2 + 1 \) gives \( 2x \).

Multiplying these derivatives together gives us the complete derivative of the composite function. Consideration of the chain rule ensures no steps are skipped and each function's inner workings are clearly spelled out. This makes differentiating even complex nested functions more approachable.