Question

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

Short answer

Answer: The derivative of the given function is \(f'(x) = \frac{a x+b}{h}\).

Step-by-step solution

Apply the constant multiple rule

\(f'(x)=\frac{1}{2h}(a x^{2}+2 b x+c)'\) Next, we will apply the power rule for differentiation on each term inside the parentheses.

Apply the power rule

The power rule states that: \((x^n)' = nx^{n-1}\). 1. Differentiate the first term \((a x^{2})'\) \((ax^2)' = 2ax\) 2. Differentiate the second term \((2bx)'\) \((2bx)' = 2b\) 3. Differentiate the third term \((c)'\) \(c' = 0\) Now, substitute these derivatives back into the function.

Substitute the derivatives back into the function

\(f'(x)=\frac{1}{2h}(2ax + 2b)\) As a final step, we can simplify our expression by canceling out common factors.

Simplify the expression

\(f'(x)=\frac{2ax + 2b}{2h} = \frac{2(a x+b)}{2 h} = \frac{a x+b}{h}\) The derivative of the given function is: \(f'(x) = \frac{a x+b}{h}\).

Key concepts

Constant Multiple Rule in Differentiation

When you encounter a function that is being multiplied by a constant, the constant multiple rule simplifies the process of finding its derivative. This rule allows you to take the constant out of the differentiation operation and multiply it by the derivative of the function. Specifically, if you have a function such as cf(x), where c is a constant and f(x) is a function of x, the derivative is c times the derivative of f(x).
Think of it like this: if you’re tasked with distributing the same number of cookies to your friends every day, the total number of cookies you hand out depends solely on the number of friends, not on the cookies themselves. The multiplication of the function by a constant works similarly when taking the derivative.
For example, in our exercise, the original function f(x) is multiplied by 1/(2h). According to the constant multiple rule, we take the derivative of the function f(x) and then multiply the result by 1/(2h).

Power Rule for Differentiation

The power rule for differentiation is one of the most fundamental tools in calculus. It applies to any function of the form x^n, where n is any real number. The rule states that the derivative of such a function is nx^{n-1}. It's like reducing the power of a term by one and then multiplying the term by its original power.
Imagine climbing a staircase, and each step you take, you go up by a certain height (power). If you took one step (reduce the power by one) and then multiplied the number of steps by the height of the original step (the original power), you'd have the effort needed to advance to that point. In a mathematical context, if we want to differentiate a x^2, we multiply 2 (the original power) with a x, and thus the derivative is 2ax.

Simplifying Expressions

After applying the rules of differentiation, you may often end up with expressions that can be simplified. This involves combining like terms, canceling out common factors, or rearranging expressions to make them more readable and easier to understand. It's much like cleaning up after a meal; by putting things away and wiping down surfaces, you make the kitchen neat and functional again. Similarly, simplifying an expression makes it clean and easy to work with.
In the context of our exercise, after differentiating and applying the constant multiple rule, simplification helps to eliminate the common factor of 2 in the numerator and denominator. By reducing the expression (2ax + 2b) / (2h), we neatly arrive at the final, simplified derivative: (ax + b) / h.