91Ó°ÊÓ

Trigonometric functions like sine and cosine are crucial in calculus and appear frequently in physics and engineering problems. When we differentiate these functions, we use specific rules that are necessary for finding their derivatives. Given a function like \( x \sin y \), we can find its partial derivatives by considering the trigonometric identities:

\[ \frac{d}{dx}(\sin x) = \cos x \]
and
\[ \frac{d}{dx}(\cos x) = -\sin x \].

These rules help us handle expressions where sine and cosine are involved. In our example, to find \( \frac{\partial z}{\partial y} \) where \( z = x \sin y \), treat \( x \) as a constant when differentiating. This results in \( x \cos y \), showing how trigonometric derivatives can be applied to functions where different variables interact.

If you have a function composed of trigonometric elements, remember these derivatives and rules to simplify the process.

Partial derivatives are a powerful tool for dealing with functions of multiple variables. They allow us to see how a function changes with respect to one variable while keeping the others constant. In our problem, the function \( z = x \sin y \) involves two variables, \( x \) and \( y \).

To find its partial derivatives:

• Differentiate \( z \) with respect to \( x \) while keeping \( y \) constant: \( \frac{\partial z}{\partial x} = \sin y \).
• Differentiate \( z \) with respect to \( y \) while keeping \( x \) constant: \( \frac{\partial z}{\partial y} = x\cos y \).

This approach allows us to understand the sensitivity of \( z \) to changes in \( x \) and \( y \) independently.

Partial derivatives are widely used in fields like optimization and economics where multi-variable functions are common. Remember, the idea is to keep all but one variable constant, which makes it easier to analyze the function’s behavior.

Theorem 12.7 offers a systematic approach to find derivatives of composite functions using the chain rule. When functions are nested or interconnected, this theorem provides a clear path to their derivatives.

For the function \( z = x \sin y \) where \( x = t^2 \) and \( y = 4t^3 \):

• We use partial derivatives: \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \).
• Calculate \( \frac{dx}{dt} \) as \( 2t \) and \( \frac{dy}{dt} \) as \( 12t^2 \).

Theorem 12.7 shows how these interact by:
\[ \frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt} \].
Substituting the values, we find the derivative in terms of \( t \). This comprehensive approach not only helps in calculus problems but applies widely to physics and any dynamic system analysis.