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
Solution:
Step 1:
Apply the differentiation operator $\frac{d}{dx}$ 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 $\frac{dy}{dx}$.
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 $\frac{d}{dx}(8) + \frac{d}{dx}(-x^3)$.
Step 3.1.2:
Acknowledge that the derivative of a constant, such as $8$, is zero, leading to $0 + \frac{d}{dx}(-x^3)$.
Step 3.2:
Focus on finding $\frac{d}{dx}(-x^3)$.
Step 3.2.1:
Recognize that the coefficient $-1$ is a constant multiplier and apply the constant multiple rule to get $-1 \cdot \frac{d}{dx}(x^3)$.
Step 3.2.2:
Apply the Power Rule for differentiation, which states that the derivative of $x^n$ is $nx^{n-1}$, where $n$ is a constant. In this case, $n=3$, so we have $-1 \cdot (3x^{3-1})$.
Step 3.2.3:
Perform the multiplication to obtain $-3x^2$.
Step 3.3:
Combine the results to complete the differentiation of the right-hand side, yielding $-3x^2$.
Step 4:
Express the derivative of $y$ with respect to $x$ as $\frac{dy}{dx} = -3x^2$.
Step 5:
Conclude the differentiation process by replacing $y$ with $\frac{dy}{dx}$ to obtain the final result $\frac{dy}{dx} = -3x^2$.
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 $\frac{dy}{dx}$.
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 $nx^{n-1}$, 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.
