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The chain rule is a crucial concept in calculus, particularly when dealing with functions of multiple variables. It comes into play when you have a composition of functions and need to find their derivatives. In the context of partial derivatives, the chain rule helps us differentiate a function with respect to one variable while considering other variables as constants.

Consider the function \( f(x, y) = \sin(2x + 5y) \). Here, the inner function is \( 2x + 5y \), and the outer function is \( \sin(u) \), where \( u = 2x + 5y \). To find the partial derivative with respect to \( x \), we apply the chain rule:

\[ \frac{\partial}{\partial x} f(x, y) = \cos(2x + 5y) \cdot \frac{\partial}{\partial x}(2x + 5y) \]

This results in multiplying the derivative of the outer function, \( \cos(2x + 5y) \), by the derivative of the inner function, \( 2 \). This yields \( f_x = 2\cos(2x + 5y) \). By applying the chain rule, we handle the complexity of differentiating composite functions smoothly.

Differentiation with respect to variables involves taking the derivative of a function while treating certain variables as constants. In the scenario of partial derivatives, you focus on how a function changes as you vary one variable at a time, keeping others fixed.

For the function \( f(x, y) = \sin(2x + 5y) \), we first differentiated with respect to \( x \) to get \( f_x = 2\cos(2x + 5y) \). The next step involves differentiating \( f_x \) with respect to \( y \). Here, we use the chain rule again:

\[ \frac{\partial}{\partial y} f_x = \frac{\partial}{\partial y} [2\cos(2x + 5y)] = -10\sin(2x + 5y) \]

The trick is to handle the expressions carefully, recognizing how each variable influences the function. By following this method, we get insight into how changes in each variable affect the function, one variable at a time.

Trigonometric functions like \( \sin \) and \( \cos \) are common in calculus because they describe periodic phenomena in mathematics. They pose unique challenges and opportunities when differentiating, especially with functions of multiple variables.

For a function like \( f(x, y) = \sin(2x + 5y) \), the derivatives involve these trig functions directly. When taking the derivative, the rules of trigonometric differentiation apply, such as:

• The derivative of \( \sin(u) \) is \( \cos(u) \)
• The derivative of \( \cos(u) \) is \(-\sin(u) \)

These rules help us compute the derivatives efficiently, as seen in our partial derivatives:
\( f_x = 2\cos(2x + 5y) \) and further differentiating with respect to \( y \) gives \( -10\sin(2x + 5y) \).

Understanding how to apply these derivatives is vital, especially in calculating partial derivatives where trigonometric functions add layers of complexity to the solution.