91Ó°ÊÓ

The chain rule is a fundamental tool in calculus used to differentiate composite functions. When we have a function within another function, like in our exercise, the chain rule allows us to differentiate them efficiently. In our example, the expression is of the form \( y = (g(x))^{-1} \), indicating a function within a reciprocal and raised to a power. To apply the chain rule:

• First, differentiate the outer function with respect to the inner function.
• Then, multiply this result by the derivative of the inner function.

For the equation, this involves:1. Differentiating the outer function, which is \( u(x) = (3x^2 + x - 3)^9 \). Here it becomes \( -u(x)^{-2} \).2. Multiply it by the derivative of the inner function \( u'(x) \). Together, they create the formula: \[D_x(y) = -u(x)^{-2} \cdot u'(x)\]This process allows us to unravel the complexities of nested functions. It simplifies the task of finding derivatives for functions that aren't straightforward.

Differentiating functions involving reciprocals requires specific attention. A reciprocal function is simply the inverse of a given function, represented as \( u(x)^{-1} \). When differentiating reciprocal functions, the power rule within the reciprocal must be considered.To differentiate a reciprocal function like \( y = \frac{1}{f(x)} \), we apply the formula:\[D_x(y) = -f(x)^{-2} \cdot f'(x)\]This means:

• The derivative changes the sign, indicated by the negative sign.
• The base function \( f(x) \) is raised to the power of \(-2\).
• Multiplication by the derivative of \( f(x) \) helps account for the reciprocal nature.

In our problem, we have \((3x^2 + x - 3)^{-1}\), becoming part of a larger power, which means careful application of these rules to ensure accuracy in finding the derivative.

The power function differentiation is well-used in calculus. It refers to finding derivatives of functions in the form \( x^n \), where \( n \) is a constant. Here is how you perform power function differentiation:

• Bring down the exponent as a coefficient in front.
• Subtract one from the original exponent.

For a function \( y = (3x^2 + x - 3)^9 \), this becomes:\[y' = 9 \cdot (3x^2 + x - 3)^{8} \cdot \{\text{derivative of the inner function}\}\]It is important that the derivative of the inner function, \( v(x) = 3x^2 + x - 3 \) is calculated separately using the rules of differentiation. Once \( v'(x) \) is found (\( 6x + 1 \)), multiplying it with the coefficient derived from the power function differentiation gives us the full derivative.This process of correctly differentiating through utilizing power rules ensures accuracy and cohesiveness throughout the derivative calculation.