Some Derivative Calculus Techniques and Examples

Derivative calculus is a branch of mathematics that deals with the study of how functions change over time or space. It involves the computation of derivatives and their characteristics, which measure the rate of change by changing the functions.

It is also the fundamental unit of calculus and it measures the responsiveness of a result to a change in its given value. For instance, the velocity of an object is the derivative of its position with respect to time, which indicates when time increases if the object’s position will change quickly.

Differentiation and integration are the two main tasks of single-variable calculus, and the basic theorem of calculus connects integration with anti-differentiation. Differentiation is the procedure used to identify derivatives and the inverse of anti-differentiation.

In this article, we will discuss the definition of derivatives, different techniques to solve the derivatives, and for a better understanding of derivatives solving various examples.

Definition of Derivative:

The derivative of a function g(x) with respect to its input variable x is defined as the limit of the difference quotient as the interval between the two points approaches zero.

g’(x) = lim_h→0 [g(x + h) – g(x)]/h

Techniques to find the derivative:

• Simple Differentiation
• The First Principle Rule
• Partially Differentiation

Simple Differentiation

In this method, derivatives are found by applying rules based on the needs of the function. For example, in the case of a multiplying condition, the product rule is employed, and if variables appear in a division, the quotient rule is applied.

Moreover, the substitution approach and the chain rule can also be used to quickly calculate the derivative of complex problems if it is challenging to discover the derivative of a long statement question.

The First Principle Rule

In this method, the binomial theorem will be applied if the power or exponent of the function is positive, and the binomial series will be used if the power is negative or in fraction form. It is a lengthy method to find the derivative of the function and it can also be used to take the derivative of the function.

Partially Differentiation

In this method take the partial derivative taking one variable constant. The approach for obtaining it is to take the derivative of one variable and use the other variable as a constant. It is also used for getting the derivative of the function if a function depends on more independent variables than one, issues can be differentiated partially.

The notation for partial differentiation is stated as. If f(x, y) is defined as then the partial derivative with respect to “x” and with respect to “y” as.

∂f/∂x = f_x = partial derivative with respect to “x”.

∂f/∂y = f_y = partial derivative with respect to “y”.

Example Section:

In this section, we’ll discuss some examples to differentiate the function with respect to the independent variable.

Example 1:

Differentiate the function below:

(x² + 4) × (x + 2)

Solution:

Step 1:Let the given value is equal to function “y”.

y = (x²+ 4) × (x+2)

Step 2:Apply the derivative on both sides with respect to “x”.

d/dx (y) = d / dx {(x²+ 4) × (x+2)}

Step 3:Apply the product rule carefully and the product rule given as.

d/dx (F(x) × G(x)) = F(x) × d/dx (G(x)) + G(x) × d/dx (F(x))

 = (x²+ 4) × d/dx (x+2) + (x+2) × d/dx (x²+ 4)

Step 4:Using the sum rule and power rule of the derivative and simplify.

= (x²+ 4) × {d/dx (x) + d/dx (2)} + (x+2) × {d/dx (x²) + d/dx (4)}

= (x²+ 4) × (1+ 0) + (x+1) × {2x + 0}

Step 5:Using the algebraic technique and simplify.

= (x²+ 4) × (1) + (x+1) × {2x}

Multiply the terms.

= (x²+ 4) + 2x² + 2x

= x²+ 4 + 2x² + 2x

= x²+ 2x² + 2x + 4

= 3x² + 2x + 4

= 3x² + 2x + 4 is the derivative of the function (x²+ 4) × (x+2).

Alternatively, a differential calculator can be used to differentiate the functions without involving into complex calculations.

Example 2:

Find the partial derivative of function f (x, y) = 3x + 7y with respect to “x” and “y”.

Solution:

Step 1:Now find the partial derivative of the given function with respect to “x” and then deal with the variable “y” as a constant.

f_x = ∂f/∂x = 3

Step 2:Similarly, find the partial derivative of the given function with respect to “y” and then deal with the variable “x” as a constant.

 f_y = ∂f/∂y = 7

Example 3:

Calculate the derivative of function “y = 5x+ 2” with the help of the first principal rule.

Solution:

Step 1:Let, the given function.

y = 5x+ 2

Step 2:Change the “y” with “y + ẟy” and “x” with “x+ẟx”in the above function.

Y+ ẟy = 5(x+ẟx) + 2

Step 3: Subtract the “step2” equation by the “step1” equation and simplify.

Y+ ẟy – y = 5(x+ẟx) + 2 – (5x+ 2)

ẟy = 5(x+ẟx) + 2 – 5x- 2

ẟy = 5(x+ẟx) – 5x

Step 4:Now “ẟx” divides on both sides.

 ẟy / ẟx = 5(x+ẟx) – 5x/ ẟx

Step 5:Apply “limit ẟx → 0”on both sides and simplify.

Lim _{ẟx → 0} ẟy / ẟx = Lim _{ẟx → 0} {5(x+ẟx) – 5x/ ẟx}

dy / dx = Lim _{ẟx → 0} {5x + 5ẟx – 5x/ ẟx }

dy / dx = Lim _{ẟx → 0} {5ẟx  / ẟx }

dy / dx = Lim _{ẟx → 0} {5}

dy / dx = 5

dy / dx = 5 is the derivative of function “y = 5x+ 2”.

Summary:

In this article, we discussed the definition of derivatives and techniques to solve them. Moreover, for a better understanding of derivatives solving different examples using sum rules and algebraic techniques.