91Ó°ÊÓ

In multivariable calculus, we extend the ideas of calculus to functions that depend on two or more variables, like the function given in our exercise: \( w = y \tan(x + 2z) \). This field of study is essential for understanding complex systems in physics, engineering, and other sciences, where multiple variables influence outcomes simultaneously.
Partial derivatives are a crucial tool in multivariable calculus. They measure how a function changes as one variable changes, while all other variables remain fixed.
For the function \( w = y \tan(x + 2z) \), multivariable calculus allows us to systematically find how changes in each variable (\(x\), \(y\), and \(z\)) affect \(w\) by computing its partial derivatives. These derivatives give us a thorough view of the function’s behavior from different angles.

The chain rule is a fundamental technique in calculus, especially when dealing with composite functions like \( \tan(x + 2z) \) in our problem. The composite function has an "inside" function \( x + 2z \) and an "outside" function \( \tan \).

• To differentiate the outer function \( \tan(x+2z) \), we find the derivative \( \sec^2(x + 2z) \).
• Then, we multiply this by the derivative of the inside function \( x + 2z \) with respect to \(z\), which is \(2\).

Using the chain rule, we compute \( \frac{\partial w}{\partial z} = y \cdot 2 \cdot \sec^2(x + 2z) \), effectively capturing the compounded effect of \(z\) on \(w\). Thus, the chain rule helps us navigate through layers of variable interactions neatly.

Trigonometric derivatives are critical when working with functions that include trigonometric terms, like the tangent function in \( w = y \tan(x + 2z) \). These derivatives allow us to understand how trigonometric functions change with respect to their inputs.

• The derivative of \( \tan(u) \) with respect to \( u \) is \( \sec^2(u) \), a key fact used to tackle the partial derivatives in this problem.
• Knowing this, we can efficiently determine the effect of \( x \) and \( z \) on the function's tangent component.

Applying trigonometric derivatives, we obtain \( \frac{\partial w}{\partial x} = y \cdot \sec^2(x + 2z) \). These derivatives make handling the curvature and wave-like nature of trigonometric functions much easier in calculus.