Find dy/dx y=-5x^3
The problem provided is an instruction to compute the derivative of a given function with respect to x. The function in question is y = -5x^3, which is a cubic polynomial in x. The derivative dy/dx represents the rate of change of the function y with respect to x. Calculating the derivative involves using the rules of differentiation from calculus to find the expression that gives the slope of the tangent line to the curve at any point x.
$y = - 5 x^{3}$
Answer
Solution:
Step 1:
Apply the differentiation operator to both sides of the equation: $\frac{d}{dx}(y) = \frac{d}{dx}(-5x^3)$.
Step 2:
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:
Recognize that $-5$ is a constant factor and can be pulled out of the differentiation operation: $-5 \cdot \frac{d}{dx}(x^3)$.
Step 3.2:
Apply the Power Rule for differentiation, which states that the derivative of $x^n$ is $n \cdot x^{n-1}$, where $n$ is a constant: $-5 \cdot (3x^{3-1})$.
Step 3.3:
Simplify the expression by multiplying the constants: $-15x^2$.
Step 4:
Combine the results to form the derivative equation: $\frac{dy}{dx} = -15x^2$.
Step 5:
Substitute $\frac{dy}{dx}$ for $y$ to express the final derivative: $\frac{dy}{dx} = -15x^2$.
Knowledge Notes:
To solve the problem of finding the derivative of a function, one must understand the following concepts:
Derivative: The derivative of a function at a point is the rate at which the function's value changes at that point. It is denoted as $\frac{dy}{dx}$ when differentiating $y$ with respect to $x$.
Differentiation Rules: There are several rules to help differentiate functions:
Constant Rule: The derivative of a constant is zero.
Power Rule: The derivative of $x^n$ with respect to $x$ is $n \cdot x^{n-1}$.
Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
Applying the Differentiation Operator: When you apply $\frac{d}{dx}$ to both sides of an equation, you differentiate each side separately.
Simplifying Expressions: After applying differentiation rules, it's important to simplify the expression to find the simplest form of the derivative.
By understanding and applying these rules, one can find the derivative of a given function with respect to a variable.
