Question

Differentiate the following funetions: \(y=a+b x+c x^{2}\)

Short answer

Question: Find the derivative of the quadratic function \(y = a + bx + cx^2\) with respect to x. Answer: The derivative of the function with respect to x is \(\frac{dy}{dx} = b + 2cx\).

Step-by-step solution

Recall the basic differentiation rules

Before finding the derivative, recall the following rules for differentiation: 1. Constant Rule: The derivative of a constant is 0. 2. Power Rule: \(\frac{d}{dx}(x^n) = nx^{n-1}\), where n is the power of x. 3. Sum Rule: If we have two functions, u(x) and v(x), then the derivative of their sum is given by \(\frac{d}{dx}(u(x) + v(x)) = \frac{du}{dx} + \frac{dv}{dx}\).

Apply the differentiation rules to the given function

Using the sum rule, we can break down the given function into three components: \(\frac{dy}{dx} = \frac{d}{dx}(a) + \frac{d}{dx}(bx) + \frac{d}{dx}(cx^2)\) Now, we can apply the respective rules to each component: 1. The first component is a constant, so its derivative is 0: \(\frac{d}{dx}(a) = 0\) 2. The next component is \(bx\). Using the power rule, we have: \(\frac{d}{dx}(bx) = b\frac{d}{dx}(x^1) = b(1)(x^{(1-1)}) = b\) 3. The last component is \(cx^2\). Applying the power rule, we get: \(\frac{d}{dx}(cx^2) = c\frac{d}{dx}(x^2) = c(2)(x^{(2-1)}) = 2cx\)

Put everything together to find the derivative

Now that we have all the individual derivatives, we can simply add them all up according to the sum rule to get the final derivative of y: \(\frac{dy}{dx} = 0 + b + 2cx\) So, the derivative of the given function, \(y = a + bx + cx^2\), with respect to x is: \(\frac{dy}{dx} = b + 2cx\)

Key concepts

Power Rule

The Power Rule is a fundamental concept in differentiation you will encounter often. If you have a term of the form \(x^n\), where \(n\) is any real number, the derivative is found using the Power Rule. Essentially, you multiply the term by the power, and then subtract one from the power.

• For example, the derivative of \(x^2\) is calculated as follows: bring down the 2 (making it \(2x\)) and then decrease the power by 1, resulting in \(x^{1}\) or simply \(x\).
• Therefore, \(\frac{d}{dx}(x^2) = 2x\). The general formula is \(\frac{d}{dx}(x^n) = nx^{n-1}\).

This process simplifies finding derivatives when dealing with polynomial terms. Remember, you multiply and subtract; it’s a straightforward two-step process which becomes second nature with practice.

Constant Rule

The Constant Rule is probably the easiest rule in differentiation. It says that the derivative of any constant term is zero. This makes sense because constants do not change, so their rate of change (which is what a derivative represents) is zero.

• If you see a standalone number without a variable, its derivative is 0.
• For instance, in the function \(y = a\) where \(a\) is constant, the derivative \(\frac{d}{dx}(a) = 0\).

So, when you differentiate any expression and come across a constant, you can immediately write down zero. This makes differentiation much faster and simpler, especially when dealing with long equations with multiple terms.

Sum Rule

The Sum Rule is essential when you differentiate functions composed of several terms. It tells you that you can find the derivative of a sum by differentiating each term individually and then adding the results together.

• For example, for a function \(y = f(x) + g(x)\), the derivative \(\frac{d}{dx}(y) = \frac{d}{dx}(f(x)) + \frac{d}{dx}(g(x))\).
• This rule applies to as many terms as present in the function, meaning you separate the differentiation task for each component.

By applying the Sum Rule, you handle each part of a complex expression, using the appropriate differentiation rules on each segment. This approach simplifies breaking down large problems into manageable pieces.