91Ó°ÊÓ

Differentiating trigonometric functions is a fundamental part of calculus, which is essential for understanding how functions change. One of the most common trigonometric functions you will encounter is the sine function, denoted as \( \sin x \). The derivative of \( \sin x \) with respect to \( x \) is \( \cos x \). This means that at any point on the sine wave, the slope is determined by the cosine of that point.

When you have a constant multiplied by a trigonometric function, like in the term \( 5\sin x \), the constant can be factored out when differentiating. Therefore, the derivative of \( 5\sin x \) is simply \( 5\cos x \). This rule is quite handy and simplifies the process of differentiation.

• Remember: The derivative of \( \sin x \) is \( \cos x \).
• If multiplied by a constant, factor out the constant. E.g., \( 5\sin x \rightarrow 5\cos x \).

The power rule is a critical aspect of calculus, used to differentiate functions that are polynomials or can be rewritten as polynomials. It's particularly useful for expressions of the form \( x^n \), where \( n \) is any real number. According to the power rule, to find the derivative, you:

• Multiply the exponent \( n \) by the coefficient of \( x^n \).
• Subtract one from the exponent.

For example, if you have \( 3x^{-1} \), applying the power rule:

• Bring down \( -1 \) to multiply by 3, giving \( 3(-1) = -3 \).
• Subtract 1 from \( -1 \) to get \( -2 \), resulting in \(-3x^{-2} \).

So, the derivative of \( 3x^{-1} \) is \( -\frac{3}{x^2} \). This concept is especially helpful in calculus as it simplifies taking derivatives of many algebraic functions.

The sum rule is a straightforward yet powerful principle in differentiation. It states that the derivative of a sum of functions is the sum of their derivatives. This means if you have a function composed of several terms added together, you can differentiate each term independently and then add those results.

For instance, consider the function \( y = \frac{3}{x} + 5\sin x \). Using the sum rule, you differentiate each piece separately:

• Apply the power rule to \( \frac{3}{x} \), rewritten as \( 3x^{-1} \), which gives \(-\frac{3}{x^2} \).
• Differentiate \( 5\sin x \) using the derivative of trigonometric functions rule, giving \( 5\cos x \).

Finally, you combine the results:

\( \frac{dy}{dx} = -\frac{3}{x^2} + 5\cos x \). This rule simplifies the differentiation process, especially for functions that are sums of simpler functions.