Explicit Function

An explicit function is a mathematical expression where the dependent variable is clearly defined in terms of the independent variable. It’s commonly written as \(y = f(x)\), meaning the value of y depends directly on x. This form makes it easy to calculate the output for any given input.

Implicit and explicit functions differ in how the dependent variable is expressed in terms of the independent variable. In some cases, simplifying an implicit function can convert it into an explicit form. Here are a few differences between implicit and explicit functions.

The derivatives of explicit functions are found using basic rules of differentiation. Explicit functions are written as y = f(x), where x and y are the input and output, respectively, and the derivative of the function is y’ = f’(x). Let us take a few examples to understand how to find derivatives of an explicit function.

Example 1: Find the derivative of explicit function \(y = 3x^3 - 5x + cos(x)\)

To differentiate \(y = 3x^3 - 5x + cos(x)\), we apply standard differentiation rules with respect to x;

\( \text{Given: } y = 3x^3 - 5x + \cos(x)\\[1em] \text{Differentiate both sides with respect to } x:\\[1em] \frac{dy}{dx} = \frac{d}{dx}(3x^3) - \frac{d}{dx}(5x) + \frac{d}{dx}(\cos(x))\\[1em] \text{Compute each derivative:}\\[1em] \frac{d}{dx}(3x^3) = 9x^2\\[1em] \frac{d}{dx}(-5x) = -5\\[1em] \frac{d}{dx}(\cos(x)) = -\sin(x)\\[1em] \text{Therefore,}\\[1em] \frac{dy}{dx} = 9x^2 - 5 - \sin(x)\\[1em] \boxed{y' = 9x^2 - 5 - \sin(x)}\\ \)

Hence, the derivative is \(\frac{dy}{dx}=9x^2-5-sin(x)\)

Example 2: Find the derivative of \(x^2y - 2y = 0\)

This is an implicit function, so we’ll use implicit differentiation.

Differentiate both sides with respect to x:

\(\frac{d}{dx} (x^2y - 2y) = \frac{d}{dx} (0)\)

Applying the product rule on x2y, we get:

\(\frac{d}{dx} (x^2y) = x^2  \frac{dy}{dx} + 2x  y\)

\(\frac{d}{dx} (-2y) = -2 \frac{dy}{dx}\)

Putting it all together, we get: 

\(x^2  \frac{dy}{dx} + 2xy - 2  \frac{dy}{dx} = 0\)

Now, combine the dydx terms:

\((x^2 - 2) \frac{dy}{dx} = -2xy\)

Now solve for dydx: 

\(\frac{dy}{dx} = \frac{-2xy}{x^2 - 2}\)

Example 3: Find the derivative of the explicit function
\(y = ln(x^2 + 1)\)

Differentiate both sides with respect to x:

\(\frac{dy}{dx}=\frac{1}{x^2+1}2x=\frac{2x}{x^2+ 1}\)

So, \(\frac{dy}{dx}=\frac{2x}{x^2+ 1}\)

Here are some of the parent and student friendly tips and tricks that would help in mastering the concept of explicit function.
 

1. Start with the “machine” mindset. Think of an explicit function as a rule machine: You put in an input (x), and it tells you exactly what the output (y) is.
   
   Example: \(y = 3x + 2\) → “Multiply by 3, then add 2.”
   
   Understanding this flow builds strong intuition.
    

2. Remember that if there is one input, there will be only one output. Every function gives only one output for each input. If you plug in the same 𝑥 twice, you must get the same 𝑦 both times — that’s what makes it a function!
    

3. Always try to visualize it. Graphing makes explicit functions click. Linear functions (y=mx+b) make straight lines. Quadratic functions (y=ax2+bx+c) make parabolas. You’ll notice how changing coefficients changes the shape. Use free tools like Desmos or GeoGebra to experiment.
    

4. Connect to real life. Link explicit functions to daily life examples:
   
   Cost = 10x+5 (buying x items with tax)
   Distance = 60t (driving 60 mph for t hours)
   
   It’s easier to remember when it means something.
    

5. Make a game of it. Play “find the rule.” Give your child pairs like (1, 4), (2, 7), (3, 10). Ask them, “What’s the rule?” They’ll notice it adds 3 each time → y=3x+1. This builds pattern recognition and algebraic thinking.

Explicit functions help in making predictions and solving problems across various fields in real life. Some of these real-life applications are listed below.

1. Household budgeting: Explicit functions help in calculating expenses. For instance, the total cost (y) can be represented as the sum of a fixed rent of $7000 and $3000 for each unit (x):
   \(y = 7000 + 3000x\).
    
2. Motion calculations in physics: Distance is modeled as \(d = rt\) in uniform motion, where d is an explicit function of time t with constant rate r.
    
3. Profit and revenue analysis in a business: The equation for profit is \(P(x) = R(x) - C(x)\). If revenue and cost are both explicit, then P(x) becomes an explicit function.
    
4. Temperature conversion: Celsius is converted into Fahrenheit using the explicit function \(F = 95C+32\).
    
5. Estimating material requirements in construction: Functions like \(y = 4x + 11\) can be used to estimate the total amount of a material (y) needed to build a platform of length x meters, where 4 is the amount needed per meter and 11 is a fixed amount used at the start.