Find dy/dx y=7x^3

Solution:

Step:1 Apply differentiation to each term of the equation $y = 7x^3$ with respect to $x$. $\frac{d}{dx}(y) = \frac{d}{dx}(7x^3)$

Step:2 The derivative of $y$ in terms of $x$ is represented as $\frac{dy}{dx}$. $\frac{dy}{dx}$

Step:3 Proceed to differentiate the term on the right-hand side.

Step:3.1 Recognize that the coefficient $7$ is a constant and can be factored out of the differentiation process. $7 \cdot \frac{d}{dx}(x^3)$

Step:3.2 Apply the Power Rule of differentiation, which suggests that the derivative of $x^n$ is $n \cdot x^{n-1}$, where $n$ is a constant. In this case, $n=3$. $7 \cdot (3x^{3-1})$

Step:3.3 Simplify the expression by multiplying the constants. $21x^{3-1}$

Step:4 Combine the results to form the derivative equation. $\frac{dy}{dx} = 21x^{3-1}$

Step:5 Simplify the exponent to finalize the derivative. $\frac{dy}{dx} = 21x^2$

Knowledge Notes:

1. Differentiation is a fundamental concept in calculus that deals with finding the rate at which a function is changing at any given point.

2. The derivative of a function $f(x)$ with respect to $x$ is denoted as $\frac{df}{dx}$ or $f'(x)$.

3. The Power Rule is a basic differentiation rule that states if $f(x) = x^n$, then the derivative of $f(x)$ with respect to $x$ is $f'(x) = n \cdot x^{n-1}$.

4. Constants are not affected by differentiation and can be factored out of the derivative operation.

5. After differentiating an equation with respect to $x$, the resulting expression represents the derivative of the original function, which can be denoted as $\frac{dy}{dx}$ if the original function was $y$.

6. Simplifying the expression after applying the differentiation rules is an important step to obtain the final derivative in its simplest form.