Find dy/dx y=(3x^2-9)^-13

This problem involves finding the derivative of a given function. The function in question is y=(3x^2-9)^-13, which is a composite function where y is raised to a negative exponent. The question asks for the derivative of y with respect to x, denoted as dy/dx, which means you need to apply differentiation rules to find the rate at which y changes as x changes. Specifically, this will involve using the chain rule, as the function has an inner function (3x^2-9) and an outer function (raising to the power of -13). The chain rule is a formula for computing the derivative of the composition of two or more functions.

$y = {((3 x^{2} - 9))}^{- 13}$

Answer

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Solution:

Step 1:

Take the derivative of both sides of the equation with respect to $x$ : $\frac{d}{d x} (y) = \frac{d}{d x} ((3 x^{2} - 9)^{- 13})$ .

Step 2:

The derivative of $y$ with respect to $x$ is denoted as $\frac{d y}{d x}$ .

Step 3:

Proceed to differentiate the right-hand side of the equation.

Step 3.1:

Apply the chain rule for differentiation, which is expressed as $\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)$ , where $f (x) = x^{- 13}$ and $g (x) = 3 x^{2} - 9$ .

Step 3.1.1:

Introduce a substitution $u = 3 x^{2} - 9$ and differentiate: $\frac{d}{d u} (u^{- 13}) \cdot \frac{d}{d x} (3 x^{2} - 9)$ .

Step 3.1.2:

Utilize the power rule of differentiation, $\frac{d}{d u} (u^{n}) = n u^{n - 1}$ , where $n = - 13$ : $- 13 u^{- 14} \cdot \frac{d}{d x} (3 x^{2} - 9)$ .

Step 3.1.3:

Substitute back $u = 3 x^{2} - 9$ : $- 13 (3 x^{2} - 9)^{- 14} \cdot \frac{d}{d x} (3 x^{2} - 9)$ .

Step 3.2:

Differentiate the expression $3 x^{2} - 9$ .

Step 3.2.1:

Apply the sum rule of differentiation to find the derivative of $3 x^{2} - 9$ : $- 13 (3 x^{2} - 9)^{- 14} (\frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 9))$ .

Step 3.2.2:

Since $3$ is a constant, differentiate $3 x^{2}$ by taking $3$ out: $- 13 (3 x^{2} - 9)^{- 14} (3 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 9))$ .

Step 3.2.3:

Apply the power rule to $x^{2}$ : $- 13 (3 x^{2} - 9)^{- 14} (3 (2 x) + \frac{d}{d x} (- 9))$ .

Step 3.2.4:

Multiply $2$ and $3$ together: $- 13 (3 x^{2} - 9)^{- 14} (6 x + \frac{d}{d x} (- 9))$ .

Step 3.2.5:

The derivative of a constant is zero: $- 13 (3 x^{2} - 9)^{- 14} (6 x + 0)$ .

Step 3.2.6:

Simplify the expression.

Step 3.2.6.1:

Combine $6 x$ and $0$ : $- 13 (3 x^{2} - 9)^{- 14} (6 x)$ .

Step 3.2.6.2:

Multiply $6$ by $- 13$ : $- 78 (3 x^{2} - 9)^{- 14} x$ .

Step 3.3:

Express the negative exponent as a reciprocal: $- 78 \frac{1}{(3 x^{2} - 9)^{14}} x$ .

Step 3.4:

Combine terms.

Step 3.4.1:

Combine $- 78$ and the reciprocal: $\frac{- 78}{(3 x^{2} - 9)^{14}} x$ .

Step 3.4.2:

Place the negative sign in front of the fraction: $- \frac{78}{(3 x^{2} - 9)^{14}} x$ .

Step 3.4.3:

Combine $x$ with the fraction: $- \frac{x \cdot 78}{(3 x^{2} - 9)^{14}}$ .

Step 3.4.4:

Position $78$ to the left of $x$ : $- \frac{78 x}{(3 x^{2} - 9)^{14}}$ .

Step 4:

Set the left side equal to the right side to reform the equation: $y = - \frac{78 x}{(3 x^{2} - 9)^{14}}$ .

Step 5:

Substitute $\frac{d y}{d x}$ for $y$ : $\frac{d y}{d x} = - \frac{78 x}{(3 x^{2} - 9)^{14}}$ .

Knowledge Notes:

The problem-solving process involves several key concepts of calculus and differentiation:

Derivative: The derivative of a function measures the rate at which the function value changes as its input changes. Notationally, the derivative of $y$ with respect to $x$ is written as $\frac{d y}{d x}$ .
Chain Rule: This is a formula for computing the derivative of the composition of two or more functions. If $f$ and $g$ are functions, then the chain rule expresses the derivative of their composition $f (g (x))$ in terms of the derivatives of $f$ and $g$ .
Power Rule: A basic rule of differentiation that states if $f (x) = x^{n}$ , then the derivative $f^{'} (x) = n x^{n - 1}$ . It applies to any real number $n$ and is used when differentiating polynomials.
Sum Rule: This rule allows for the differentiation of a function that is the sum of two or more functions. The derivative of a sum is equal to the sum of the derivatives.
Constants: The derivative of a constant is zero. This is because a constant does not change, so its rate of change is zero.
Negative Exponent Rule: This rule states that for any nonzero number $b$ and any integer $n$ , $b^{- n} = \frac{1}{b^{n}}$ . This is used to simplify expressions with negative exponents.

In the given problem, these rules are applied in sequence to find the derivative of the function $y = (3 x^{2} - 9)^{- 13}$ with respect to $x$ . The final result is expressed as a derivative $\frac{d y}{d x}$ , which is the solution to the problem.