91Ó°ÊÓ

The constant multiplier rule is a fundamental principle in calculus. It helps simplify differentiation when a constant is involved.
This rule states: When you have a constant multiplied by a function, you can take the derivative of the function first and then simply multiply by the constant.
Notation: If you have a function of the form \( k \times f(x) \), where \( k \) is a constant, the derivative is given by \( k \times f'(x) \).
In our problem, the constant is \( \sqrt{2} \). So, instead of differentiating the entire expression, we first handle the derivative of the trigonometric part and multiply the result by \( \sqrt{2} \).
Example simplify: For function \( s(x) = \sqrt{2} (3 \cos x - 2 \sin x)\), first define \( g(x) = 3 \cos x - 2 \sin x \), and then apply the rule after finding \(g'(x) \).

Understanding derivatives of trigonometric functions is essential. The basic derivatives you need to know:

• The derivative of \( \cos x \) is \( -\sin x \).
• The derivative of \( \sin x \) is \( \cos x \).

In our problem, the function \( g(x) = 3 \cos x - 2 \sin x \) consists of standard trigonometric functions.
When we differentiate sign and cosign, we get:
\( g'(x) = 3(- \sin x) - 2( \cos x ) = -3 \sin x - 2 \cos x \).
Cordless also keep the coefficients and multiply them accordingly.

Differentiation is a core concept in calculus, and knowing the rules simplifies the process.
Important rules to remember:

• Constant rule: Derivative of a constant is 0.
• Power rule: For a function \( x^n \), its derivative is \ nx^{n-1} \.
• Sum rule: \( (f + g)' = f' + g' \), can be used for combining derivatives of two functions.
• Constant multiplier rule: \ (k \times f(x))' = k \times f'(x) \.

Combining these rules allows us to find complex derivatives efficiently.
In this example, we applied the constant multiplier and sum rules to solve the given function \( s(x) = \sqrt{2} (3 \cos x - 2 \sin x) \).
By breaking down the problem step-by-step, we ensure all concepts and differentiation rules are properly applied to get the final result.