Find dy/dx y=2x^3-2x

Solution:

Step 1

Apply the differentiation operator $\frac{d}{dx}$ to both sides of the given function $y = 2x^3 - 2x$.

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

Utilize the Sum Rule in differentiation to separate the terms, giving $\frac{d}{dx}(2x^3) + \frac{d}{dx}(-2x)$.

Step 3.2

Find the derivative of the first term $\frac{d}{dx}(2x^3)$.

Step 3.2.1

Recognize that $2$ is a constant factor and can be pulled out of the derivative, yielding $2\frac{d}{dx}(x^3)$.

Step 3.2.2

Apply the Power Rule, which states that the derivative of $x^n$ is $nx^{n-1}$, where $n$ is a constant, to obtain $2(3x^2)$.

Step 3.2.3

Simplify the expression by multiplying the constants to get $6x^2$.

Step 3.3

Now, differentiate the second term $\frac{d}{dx}(-2x)$.

Step 3.3.1

As before, $-2$ is a constant and comes out of the derivative, resulting in $-2\frac{d}{dx}(x)$.

Step 3.3.2

Again, apply the Power Rule for $n=1$ to find the derivative of $x$, which is $1$, giving $-2 \cdot 1$.

Step 3.3.3

The expression simplifies to $-2$ after multiplying the constants.

Step 4

Combine the results of the derivatives to form the complete derivative of the function.

Step 5

Finally, express the derivative of $y$ with respect to $x$ as $\frac{dy}{dx} = 6x^2 - 2$.

Knowledge Notes:

1. Differentiation: Differentiation is the process of finding the derivative of a function, which represents the rate at which the function value changes with respect to changes in its input value.

2. Derivative Notation: The derivative of a function $y$ with respect to a variable $x$ is denoted as $\frac{dy}{dx}$ or $y'$.

3. 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. Mathematically, $\frac{d}{dx}(f + g) = \frac{df}{dx} + \frac{dg}{dx}$.

4. Constant Factor Rule: If a term in the function is multiplied by a constant, the derivative of that term is the constant multiplied by the derivative of the term. The rule is expressed as $\frac{d}{dx}(cf) = c\frac{df}{dx}$, where $c$ is a constant.

5. Power Rule: The Power Rule is a basic rule of differentiation that applies to functions of the form $x^n$, where $n$ is any real number. The rule states that $\frac{d}{dx}(x^n) = nx^{n-1}$.

6. Combining Derivatives: After finding the derivatives of individual terms, they can be combined according to the original equation to find the overall derivative of the function.

7. Simplification: After applying the differentiation rules, the resulting terms are often simplified by performing arithmetic operations such as multiplication or addition.