Find dy/dx y=80/(x^9)

Solution:

Step 1:

Take the derivative of both sides of the equation with respect to $x$: $\frac{d}{dx}(y) = \frac{d}{dx}\left(\frac{80}{x^9}\right)$.

Step 2:

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

Step 3:

Proceed to find the derivative of the right-hand side of the equation.

Step 3.1:

Recognize that $80$ is a constant factor and can be pulled out of the derivative: $\frac{d}{dx}\left(\frac{80}{x^9}\right) = 80\frac{d}{dx}\left(\frac{1}{x^9}\right)$.

Step 3.2:

Utilize the rules for exponents.

Step 3.2.1:

Express $\frac{1}{x^9}$ as $x^{-9}$: $80\frac{d}{dx}(x^{-9})$.

Step 3.2.2:

Apply the exponent multiplication rule.

Step 3.2.2.1:

Invoke the power rule for derivatives, which states that $\frac{d}{dx}(a^{mn}) = a^{mn}$: $80\frac{d}{dx}(x^{-9})$.

Step 3.2.2.2:

Perform the multiplication of the exponents: $80\frac{d}{dx}(x^{-9})$.

Step 3.3:

Apply the Power Rule for differentiation, which states that the derivative of $x^n$ is $nx^{n-1}$, where in this case, $n$ is $-9$: $80(-9x^{-10})$.

Step 3.4:

Multiply the constant $-9$ by $80$: $-720x^{-10}$.

Step 3.5:

Simplify the expression.

Step 3.5.1:

Rewrite the expression using the rule for negative exponents: $b^{-n} = \frac{1}{b^n}$: $-720\frac{1}{x^{10}}$.

Step 3.5.2:

Combine the constant and the term.

Step 3.5.2.1:

Combine $-720$ and $\frac{1}{x^{10}}$ to form $\frac{-720}{x^{10}}$.

Step 3.5.2.2:

Place the negative sign in front of the fraction: $-\frac{720}{x^{10}}$.

Step 4:

Formulate the final equation by equating the left-hand side with the right-hand side: $y = -\frac{720}{x^{10}}$.

Step 5:

Substitute $\frac{dy}{dx}$ for $y$ to express the derivative: $\frac{dy}{dx} = -\frac{720}{x^{10}}$.

Knowledge Notes:

The problem involves finding the derivative of a function with respect to $x$. The function is given as $y = \frac{80}{x^9}$. The process of finding the derivative is known as differentiation.

1. Differentiation: This is the process of finding the derivative of a function. The derivative represents the rate at which a function is changing at any point and is a fundamental concept in calculus.

2. Constant Multiplication Rule: When differentiating a function that is multiplied by a constant, the constant can be pulled out of the differentiation operation.

3. Power Rule: This is a basic rule of differentiation that states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. This rule is applied to functions where the variable $x$ is raised to a power $n$.

4. Negative Exponents: A term with a negative exponent can be rewritten as the reciprocal of the term with a positive exponent. For example, $x^{-n} = \frac{1}{x^n}$.

5. Combining Terms: After differentiating and applying rules, terms are combined to simplify the expression to its simplest form.

By applying these rules and steps, the derivative of the given function with respect to $x$ is found to be $\frac{dy}{dx} = -\frac{720}{x^{10}}$.