Differentiation Rules: Trigonometric & Inverse Trigonometric Function

Differentiation rules allow us to evaluate the derivatives of specific functions. The attribute of linearity is crucial for the procedure of differentiation or finding the derivative of a function. For functions produced from the fundamental elementary functions using the methods of addition and multiplication by a constant integer, this characteristic makes the derivative more natural.

Key Takeaways: Differentiation rules, Trigonometric, Inverse Trigonometric, Functions, Addition, Division, Multiplication, Division

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What are Differentiation Rules?

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The derivatives should be evaluated correctly. In fact, differentiating a function helps to provide a universal outcome. As a result, you must comprehend and observe some differentiation concepts or rules. Some Important rules of differentiation include power rule, product rule, chain rule, quotient rule and difference rule. These rules are also sometimes known as derivative rules.

Differentiation Rules

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Continuity and Differentiability Detailed Video Explanation:

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Rules of Differentiation

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Some of the important rules of differentiation are:

Power Rule

This is a primary differentiating rule and it is very easy to understand . Let’s see how a function changes when a power method is implemented.

If f(x) = xⁿ,

Then, f’(x) = d/dx (xⁿ) = nx^n-1

Let’s look at an example, then it will make you understand the application properly.

If f(x) = x⁶

Then, d/dx (x⁶) = 6x^6-1 = 6x⁵

Sum Rule

The sum rule of differentiation suggests the following changes when a function is expressed by the difference or methods of integration.

If f(x) = m(x) ± n(x)

Then, f’(x) = m’(x) ± n’(x)

This formula suggests that the smaller functions’ signs will be preserved, but these functions will adhere to the derivative rules. Consider the following example.

If f(x) = x² + x³

Then, f’(x) = 2x + 3x²

This is how the sum rule of derivatives is implemented. 

Product Rule

This rule states that if a variable's function is the product of two other functions, then the result will be as follows.

If f(x) = m(x) × n(x),

Then, f’(x) = m′(x) × n(x) + m(x) × n′(x)

Look at this example to get a better understanding of this rule, 

If f(x) = x² × x³

Then, f’(x) = d/dx (x² × x³)

= x³ × d/dx (x²) + x² × d/dx (x³)

= x³ × 2x + x² × 3x²

= 2x⁴ + 3x⁴

= 5x4

Quotient Rule

The quotient rule derivatives indicate how to differentiate a function with two terms in division mode. The rule suggests the following.

If f(x) = m(x) / n(x),

Then, f′(x) = m’(x) × n(x) − m(x) × n’(x) / (n(x))²

If you follow the rule and enter the values of the functions after the differential equation, you will get the required answer.

Chain Rule

When a function is expressed by a function with another variable, and this function is expressed by the first function's variable, the chain rule suggests the differential operation below.

If f(x) = m(u) and u = n(x),

Then, f’(x) = d/dx f(x) = d/du m(u) × d/dx n(x)

This formula or rule is relatively simple to apply if you pay attention to the radio waves in each stage and understand how to prove the chain rule.

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Differentiation Rules for Trigonometric Functions

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There are about six trigonometric functions such as, sin, cos, tan, csc, sec, and cot. Here are the given rules or formulas to help you determine the derivatives of trigonometric functions.

• The derivative of sin x is, d/dx (sin x) = cos x
• The derivative of cos x is, d/dx (cos x) = - sin x
• The derivative of tan x is, d/dx (tan x) = sec²x
• The derivative of csc x is, d/dx (csc x) = - csc x cot x
• The derivative of sec x is, d/dx (sec x) = sec x tan x
• The derivative of cot x is, d/dx (cot x) = - csc²x

Also Read: Quadratic Equations Formula

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Differentiation Rules for Inverse Trigonometric Functions

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To the aforementioned trigonometric functions, there exist 6 inverse trigonometric functions. Remember that there are no trigonometric/inverse trigonometric functions involved in the derivatives of inverse trigonometric functions. The rules are as follows:

• The derivative of inverse sine is, d/dx (sin^-1x) = 1/√(1-x²)
• The derivative of inverse cosine is, d/dx (cos^-1x) = -1/√(1-x²)
• The derivative of inverse tan is, d/dx (tan^-1x) = 1/(1 + x²)
• The derivative of inverse cosec is, d/dx (csc^-1x) = -1/ [x √(x² - 1) ], x ≠ 1, -1, 0
• The derivative of inverse sec is, d/dx (sec^-1x) = 1/ [x √(x² - 1) ], x ≠ 1, -1, 0
• The derivative of inverse cot is, d/dx (cot^-1x) = -1/(1 + x²)

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Things to Remember

Following are some important points:

• The chain rule has been known since the end of the 17th century, when Isaac Newton and Leibniz first unearthed calculus.
• Calculations involving computing the derivatives of complicated expressions, such as those encountered in many physics applications, are made easier with this rule.
• We can find rates of change using differentiation. It enables us to determine the rate of change of velocity with respect to time, for example.
• Differentiation rules help us to evaluate the derivatives of specific functions. 
• Some of the Important rules of differentiation are power rule, product rule, chain rule, quotient rule and difference rule.

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