Question

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=4 x-3 x^{3} $$

Short answer

The derivative of the function \(y=4x-3x^3\) is \(y'=4-9x^2\).

Step-by-step solution

Apply Constant Rule

From the constant rule, let's isolate the first term. The coefficient 4 in front of \(x\) is a constant. Thus the derivative of \(4x\) w.r.t \(x\) equals the constant, which in this case, is 4.

Apply Power Rule

For the second term, \(3x^3\), we apply the power rule. We get -3 multiplied by the derivative of \(x^3\). The power rule states that the derivative of \(x^3\) is 3 times the original power of \(x^{3-1}\), hence it gives -3 * 3 * \(x^{3-1}\) which will be equal to -9\(x^2\).

Sum the results

After finding the derivative for both terms, we now have to sum the results. The derivative of the first term which is 4 and the derivative of the second term which is -9\(x^2\) are added together to form the derivative for the entire function, which is \(y'\) = 4 - 9\(x^2\).

Key concepts

Derivative of a Function

Understanding the derivative of a function is akin to taking a glimpse into the heart of calculus. It gives us the rate at which one quantity changes with respect to another. In practical terms, if you consider the function as a formula that describes a physical phenomenon, the derivative tells you how fast the outcome of this formula changes when you adjust the input values.

For example, if we think of our function as an expression of distance over time, the derivative would provide us with the speed because it indicates how quickly the distance changes as time advances. In mathematical language, if we have a function denoted by y=f(x), the derivative of y with respect to x is often written as f'(x) or \( \frac{dy}{dx} \). It is the 'prime' notation or the 'dy over dx' notation respectively, both signifying the concept of differentiation with respect to x.

Constant Rule

The constant rule is an essential concept for anyone embarking on the study of calculus. It simply states that the derivative of a constant is zero. Why? Because a constant doesn’t change, and the derivative measures how much something is changing. It’s straightforward: if something never changes, no matter how you alter any other part of the equation, its rate of change is nil.

Applying the Constant Rule

In practice, when you come across a term like \(4x\), the \(4\) is treated as a constant multiplier of \(x\), which means its presence doesn't affect the derivative of \(x\) itself. According to the constant rule, when differentiating \(4x\), we keep the \(4\) as it is and simply differentiate \(x\), which gives us \(4 \times 1\) since the derivative of \(x\) with respect to \(x\) is \(1\).

When solving problems, remember this rule to streamline the process: Constant factors can be pulled out of the derivative and whatever remains - which is variable - will be subjected to differentiation.

Power Rule

When dealing with polynomials or any terms raised to a power, the power rule of differentiation is your mathematical lifeline. It directly addresses the rate of change for expressions where the variable is raised to a non-zero real number. The rule is beautifully succinct and states that to differentiate \(x^n\), where \(n\) is a real number, you multiply the entire term by the exponent \(n\) and subtract one from that exponent to create the new power of \(x\).

For the uninitiated, the formula looks like this: \((\frac{d}{dx}[x^n] = n x^{n-1}\)).

Putting the Power Rule into Action

Consider the term \(3x^3\). To find the derivative, we take the exponent \(3\) and bring it in front of the \(x\), and reduce the exponent by one. Now we have \(3 \times 3x^{3-1}\), which simplifies to \(9x^2\).

If you're following the exercise provided, the power rule makes short work of finding the derivative of the term with a power. It's a straightforward rule that aids in tackling seemingly complex differentiation problems with confidence.