91Ó°ÊÓ

The Power Rule is one of the most fundamental differentiation rules for finding the derivative of a power function. This rule is incredibly useful for polynomials or any function where you have a variable raised to a power. In general, if you have a function written in the form of \( x^n \), its derivative is given by \( nx^{n-1} \). This means you multiply by the power and then reduce the power by one.

When dealing with functions that include fractions, like \( \frac{5}{x^2} \), we rewrite them as \( 5x^{-2} \) to easily apply the power rule. Here is how it works:

• Take the function \( 5x^{-2} \). The power is \(-2\).
• Multiply the coefficient \(5\) by \(-2\), giving \(-10\).
• Subtract one from the power: \(-2 - 1 = -3\).

Hence, the derivative is \(-10x^{-3}\). With the Power Rule, we can quickly and effectively differentiate power functions, including both standard polynomials and those written in forms like inverse powers or roots.

Trigonometric functions have their own set of differentiation rules, just like algebraic functions. The sine function, written \( \sin x \), is a fundamental trigonometric function that we often differentiate. The derivative of \( \sin x \) is \( \cos x \). This rule is straightforward and directly applied in problems involving trigonometric terms.

In our example of differentiating the function \( f(x) = 5x^{-2} + \sin x \), to find the derivative of the \( \sin x \) term, we apply the trigonometric derivative rule:

• Differentiate \( \sin x \) to get \( \cos x \).

Trigonometric derivatives are key in calculus, especially when dealing with periodic functions or combining them with algebraic expressions. They follow defined patterns which help in simplifying more complex expressions involving sine, cosine, tangent, and other trigonometric functions.

Differentiation is a crucial concept in calculus that involves finding the derivative of a function. The derivative tells us the function's rate of change or its slope at a particular point. There are different rules for differentiating various types of functions, and knowing which rule to apply is essential.

Some important differentiation rules include:

• The Power Rule: This is used for any function that is a power of \( x \), as discussed earlier.


• Trigonometrics: Each standard trig function has a straightforward derivative. For example, \( \sin x \) becomes \( \cos x \), \( \cos x \) becomes \(-\sin x \), and \( \tan x \) is \( \sec^2 x \).


• Constants: The derivative of any constant is zero.

By combining these differentiation rules, we can solve complex differentiation problems by addressing each part of the function separately. In particular, mastering these rules allows us to handle functions that mix algebraic terms with trigonometric ones, like the original example \( f(x) = 5x^{-2} + \sin x \). This way, we find each term's derivative and then combine the results to get the final answer, \( f'(x) = -10x^{-3} + \cos x \). Remember to always identify the form of each term in your function to choose and apply the correct rule efficiently.