Find dy/dx y=8-x^3

The given problem is asking for the derivative of the function y with respect to x. To find dy/dx when y=8-x^3, one must apply the rules of differentiation to the expression 8-x^3 with respect to the variable x. This requires understanding the concept of derivatives from calculus, which are essentially the rates at which one quantity changes with respect to another.

$y = 8 - x^{3}$

Answer

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Solution:

Step 1:

Apply the differentiation operator $\frac{d}{d x}$ to both sides of the equation $y = 8 - x^{3}$ .

Step 2:

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

Step 3:

Proceed to differentiate the expression on the right-hand side.

Step 3.1:

Begin differentiation.

Step 3.1.1:

Utilize the Sum Rule in differentiation, which allows us to differentiate $8 - x^{3}$ term by term as $\frac{d}{d x} (8) + \frac{d}{d x} (- x^{3})$ .

Step 3.1.2:

Acknowledge that the derivative of a constant, such as $8$ , is zero, leading to $0 + \frac{d}{d x} (- x^{3})$ .

Step 3.2:

Focus on finding $\frac{d}{d x} (- x^{3})$ .

Step 3.2.1:

Recognize that the coefficient $- 1$ is a constant multiplier and apply the constant multiple rule to get $- 1 \cdot \frac{d}{d x} (x^{3})$ .

Step 3.2.2:

Apply the Power Rule for differentiation, which states that the derivative of $x^{n}$ is $n x^{n - 1}$ , where $n$ is a constant. In this case, $n = 3$ , so we have $- 1 \cdot (3 x^{3 - 1})$ .

Step 3.2.3:

Perform the multiplication to obtain $- 3 x^{2}$ .

Step 3.3:

Combine the results to complete the differentiation of the right-hand side, yielding $- 3 x^{2}$ .

Step 4:

Express the derivative of $y$ with respect to $x$ as $\frac{d y}{d x} = - 3 x^{2}$ .

Step 5:

Conclude the differentiation process by replacing $y$ with $\frac{d y}{d x}$ to obtain the final result $\frac{d y}{d x} = - 3 x^{2}$ .

Knowledge Notes:

Differentiation: Differentiation is the process of finding the derivative of a function, which represents the rate at which the function is changing at any given point.
Derivative Notation: The derivative of a function $y$ with respect to a variable $x$ is denoted as $\frac{d y}{d x}$ .
Sum Rule: The Sum Rule in differentiation states that the derivative of a sum of functions is the sum of the derivatives of those functions.
Constant Rule: The derivative of a constant is zero because constants do not change.
Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
Power Rule: The Power Rule states that the derivative of $x^{n}$ with respect to $x$ is $n x^{n - 1}$ , where $n$ is a constant.
Combining Rules: When differentiating expressions, we often need to combine several rules of differentiation to find the derivative of more complex expressions.