Derivada de una suma o resta de funciones

How to derive additions and subtractions of functions?

In the world of calculus, one of the most crucial and commonly used concepts is the derivation of functions. Today we are going to focus on how the rules of addition and subtraction apply when we derive functions, helping to form that mathematical foundation that will allow us to tackle more complex problems. The rule is simple: when you have an addition or subtraction of functions, the derivation is done term by term. Let's see it in detail!

Basic example: derivative of a simple sum

Let's consider the function ( y = x^3 - 2 ). There are two types of functions here: a power function ( x^3 ) and a constant function ( -2 ). Let us see how they are derived:

y' = \frac{d}{dx}(x^3) - \frac{d}{dx}(2) = 3x^2 - 0 = 3x^2.

As you can see, the derivative of a constant is 0. This transforms the derivative of the original function into ( 3x^2 ).

What happens with more terms?

Let's look at a more complex example: ( y = x^5 - x^2 + x ). Here we find three power functions. We derive each of them.

y' = 5x^4 - 2x + 1

• For ( x^5 ), we apply the power rule: ( 5x^4 ).
• For ( x^2 ), we do the same: ( 2x ).
• Finally, for ( x ), which is equal to ( x^1 ), the derivative is simply 1.

Derivatives of functions with negative exponents

Let's work with a function with negative exponents: ( y = \frac{1}{x} + x^{-2} ). Here we rewrite the functions to facilitate the derivation:

y' = -1x^{-2} - 2x^{-3}

We then perform a simplification to avoid negative exponents:

y' = -frac{1}{x^2} - \frac{2}{x^3}

Derivatives of roots and constant multiples

Now things get more interesting with roots and multiples. Consider ( y = \sqrt{x} - 3 \sqrt[3]{x} ). The key is to rewrite the function in terms of fractional powers:

y = x^{1/2} - 3x^{1/3}

By deriving:

y' = \frac{1}{2}x^{-1/2} - 1x^{-2/3}

Finally, to express this without fractions:

y' = \frac{1}{2}{2}sqrt{x}} - \frac{1}{2sqrt[3]{x^2}}}

How to practice these rules?

Practicing these techniques is essential to master function derivation. I suggest you apply the following key rules:

• Constant function: the derivative is always 0.
• Constant multiple function: multiply the derivative of the function by the constant coefficient.
• Power function: lower the exponent, reduce by one.
• Addition or subtraction: derive each term separately.

Take your time to familiarize yourself with these rules and you will see how your mathematical skills will expand incredibly. Don't forget to share your experiences and doubts in the community. We are here to support each other, go ahead and succeed in your exercises!