Question

In Exercises, find \(d y / d x\) $$ 3 x^{2}-y=8 x $$

Short answer

After applying implicit differentiation and solving for \(y'\), we find that the derivative of \(y\) with respect to \(x\) is \(y' = 6x - 8\)

Step-by-step solution

Differentiate the Equation

Differentiate the whole equation \(3 x^{2}-y=8 x\) with respect to x. Using the power rule for \(3x^{2}\) and \(8x\), we get \(6x\) and \(8\) respectively. For \(-y\), since y is a function of x, we apply the chain rule and get \(-y'\). So the differentiated equation is \(6x - y' = 8\).

Solve for \(y'\)

Rearrange the equation from Step 1 in terms of \(y'\). This involves simply isolating \(y'\) on one side of the equation. So, the final equation is \(y' = 6x - 8\)

Key concepts

Derivatives

In differential calculus, a derivative represents the rate at which a function is changing at any given point. Essentially, it's a way to measure how one quantity changes when another quantity changes. In the context of the exercise provided, we were tasked with finding \(\frac{dy}{dx}\), which is the derivative of \(y\) with respect to \(x\).
The derivative of a function is often denoted by \(y'\) or \(f'(x)\). To find it, we differentiate the function in question, which involves using various rules such as the power rule or the chain rule, depending on the complexity of the function involved.
Understanding derivatives helps in understanding how functions behave and change - integral for fields such as physics, engineering, and economics.

Chain Rule

The chain rule is a fundamental technique in calculus used for differentiating composite functions. For instance, if you have a function \(y\) that depends on \(x\), and it's expressed as some other function \(y = g(f(x))\), the chain rule helps find its derivative.
The formula for the chain rule is: \(\frac{dy}{dx} = \frac{dg}{df} \times \frac{df}{dx}\). This means you first differentiate the outer function \(g\) with respect to the inner function \(f\), and then multiply by the derivative of the inner function \(f\) with respect to \(x\).
In the exercise, when differentiating \(-y\) with respect to \(x\), we used the chain rule because \(y\) is implicitly defined as a function of \(x\). Hence, by using the chain rule, we obtained \(-y'\).

Power Rule

The power rule is one of the simplest and most frequently used rules in calculus when finding derivatives. It’s a shortcut that saves time and effort when dealing with functions that are power expressions of \(x\).
The power rule states that for a function \(f(x) = x^n\), its derivative is given by \(f'(x) = nx^{n-1}\). Simply put, you bring down the exponent as a coefficient and reduce the exponent by one.
In the provided exercise, the power rule allows us to differentiate terms like \(3x^2\) resulting in \(6x\), and it straightforwardly addresses terms like \(8x\) which differentiates to \(8\). Understanding and applying the power rule efficiently enables quick and accurate computation of derivatives in algebraic expressions.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to solve for a specific variable. After differentiating an equation, we often use algebraic skills to isolate the desired derivative, such as \(y'\).
During the exercise, once we differentiated the equation to \(6x - y' = 8\), we performed algebraic manipulation to solve for \(y'\). This involved moving all the terms related to \(y'\) to one side of the equation, resulting in \(y' = 6x - 8\).
Mastering algebraic manipulation is essential in calculus as it allows the learner to make logical deductions, simplify complex expressions, and ultimately solve for unknowns in mathematical equations.