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

Solution:

Step 1:

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

Step 2:

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

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 $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$, where $f(x) = x^{-13}$ and $g(x) = 3x^2 - 9$.

Step 3.1.1:

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

Step 3.1.2:

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

Step 3.1.3:

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

Step 3.2:

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

Step 3.2.1:

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

Step 3.2.2:

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

Step 3.2.3:

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

Step 3.2.4:

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

Step 3.2.5:

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

Step 3.2.6:

Simplify the expression.

Step 3.2.6.1:

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

Step 3.2.6.2:

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

Step 3.3:

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

Step 3.4:

Combine terms.

Step 3.4.1:

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

Step 3.4.2:

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

Step 3.4.3:

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

Step 3.4.4:

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

Step 4:

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

Step 5:

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

Knowledge Notes:

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

1. 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 $\frac{dy}{dx}$.

2. 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 $g$.

3. Power Rule: A basic rule of differentiation that states if $f(x) = x^n$, then the derivative $f'(x) = nx^{n-1}$. It applies to any real number $n$ and is used when differentiating polynomials.

4. 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.

5. Constants: The derivative of a constant is zero. This is because a constant does not change, so its rate of change is zero.

6. 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 $y = (3x^2 - 9)^{-13}$ with respect to $x$. The final result is expressed as a derivative $\frac{dy}{dx}$, which is the solution to the problem.