Problem 25 - Calculus Single Variable

Question

Use the Reciprocal Rule to compute the derivative of the given expression with respect to \(x\) $$ 9 /\left(3 \cos (x)+x^{3}\right) $$

Step-by-Step Solution

Answer

The derivative is \( \frac{9(3\sin(x) - 3x^2)}{(3\cos(x) + x^3)^2} \).

Step 1: Identify the expression form

The given expression is \( \frac{9}{3\cos(x) + x^3} \). This expression is in the form \( \frac{1}{u(x)} \) where \( u(x) = 3\cos(x) + x^3 \).

Step 2: Apply the Reciprocal Rule formula

For a function \( f(x) = \frac{1}{u(x)} \), the derivative \( f'(x) \) is \( -\frac{u'(x)}{u(x)^2} \). In this case, multiply by the constant 9 to get: \( f'(x) = -9 \cdot \frac{u'(x)}{u(x)^2} \).

Step 3: Differentiate the denominator

Differentiate \( u(x) = 3\cos(x) + x^3 \) with respect to \( x \). The derivative of \( 3\cos(x) \) is \(-3\sin(x) \) and the derivative of \( x^3 \) is \( 3x^2 \), thus \( u'(x) = -3\sin(x) + 3x^2 \).

Step 4: Substitute into the derivative formula

Substitute \( u(x) = 3\cos(x) + x^3 \) and \( u'(x) = -3\sin(x) + 3x^2 \) into the formula \( f'(x) = -9 \cdot \frac{u'(x)}{u(x)^2} \) to get: \[f'(x) = -9 \cdot \frac{-3\sin(x) + 3x^2}{(3\cos(x) + x^3)^2}\]

Step 5: Simplify the expression if necessary

Since the result is already simplified, the final derivative of the given expression is: \[f'(x) = \frac{9(3\sin(x) - 3x^2)}{(3\cos(x) + x^3)^2}\]

Key Concepts

Reciprocal Rule

The reciprocal rule is a handy tool in calculus for differentiating functions of the form \( \frac{1}{u(x)} \). This rule simplifies the process of finding the derivative when you have a function expressed as a fraction where the numerator is a constant or another function. The basic formula for differentiating \( f(x) = \frac{1}{u(x)} \) is:\[ f'(x) = -\frac{u'(x)}{u(x)^2} \]The negative sign shows the inverse relationship. If the original function's denominator grows, the derivative decreases, and vice versa. You're essentially "flipping" the function, hence the need for the negative sign. In our exercise, the constant 9 remains and is multiplied by the derivative of the inner function, resulting in:

• For \( f(x) = \frac{9}{3\cos(x) + x^3} \), the function \( u(x) = 3\cos(x) + x^3 \).
• Applying the reciprocal rule formula, we find \( f'(x) = -9 \cdot \frac{u'(x)}{u(x)^2} \).
• This demonstrates how constants are handled in the reciprocal rule, where they are simply multiplied into the formula.

To use the reciprocal rule effectively, ensure you can identify u(x), differentiate it orderly, and apply the rule confidently.

Trigonometric Differentiation

Trigonometric differentiation involves finding the derivative of trigonometric functions like sin, cos, and tan. In calculus, these derivatives are essential because trigonometric functions show up frequently in various applications.For cos(x) and sin(x):

• The derivative of \( \cos(x) \) is \( -\sin(x) \).
• The derivative of \( \sin(x) \) is \( \cos(x) \).

In our exercise, trigonometric differentiation appears when finding the derivative of \( u(x) = 3\cos(x) + x^3 \). Here, we employed:

• The derivative of \( 3\cos(x) \) becoming \( -3\sin(x) \).
• Combining this with the derivative of \( x^3 \), which is \( 3x^2 \).

The key is remembering the standard derivatives of trigonometric functions, then applying them within the context of your specific problem. Practice these derivatives until they become second nature.

Chain Rule

The chain rule is a fundamental concept when dealing with composite functions, meaning a function inside another function, like \( f(g(x)) \). It helps in differentiating such functions by addressing each layer of the function separately.The formula for the chain rule is:\[ \frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x) \]This rule essentially involves taking the derivative of the outer function at the inner function and then multiplying it by the derivative of the inner function.In our reciprocal rule exercise, the chain rule is indirectly part of the process since we differentiate a function contained within another function:

• Recognizing that \( u(x) = 3\cos(x) + x^3 \) can be treated by taking its components separately.
• Applying the appropriate derivatives, you are effectively seeing the chain rule in action, though not labeled explicitly.

Understanding the chain rule is crucial because it opens the door to differentiating complex functions, making calculus a more flexible tool in problem-solving.