Question

Given \(z=(x+y)^{5}, y=\sin 10 x\), find \(d z / d x\).

Short answer

\( \frac{dz}{dx} = 50(x + \sin(10x))^4 \cos(10x) \)

Step-by-step solution

Understand the composite function

Given two functions: 1) Explicitly we have: \(z=(x+y)^5\)2) And \(y=\sin(10x)\) is another function dependent on \(x\).The goal is to find the derivative of \(z\) with respect to \(x\) using the chain rule.

Apply the chain rule

The chain rule states that: \[ \frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx} + \frac{dz}{dx} \]. Since \(z\) is dependent on \(y\) and \(y\) is dependent on \(x\), apply the chain rule accordingly.

Differentiate \(z\) with respect to \(y\)

To find \( \frac{dz}{dy} \) when \(z = (x + y)^5 \): \[ \frac{dz}{dy} = 5(x + y)^4 \]

Differentiate \(y\) with respect to \(x\)

Given \(y = \sin(10x)\), the derivative of \(y\) with respect to \(x\) is: \[ \frac{dy}{dx} = 10 \cos(10x) \]

Substitute and combine the derivatives

Use the chain rule to combine the derivatives. Substituting \( \frac{dz}{dy} \) and \( \frac{dy}{dx} \) into the chain rule formula: \[ \frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx} = 5(x + \sin(10x))^4 \cdot 10 \cos(10x) \]

Key concepts

Composite Function

A composite function is a function made up of two other functions. In this exercise, we have two such functions. First, we have \(z = (x + y)^5\). This tells us that \(z\) depends on \(x + y\).
Second, we have \(y = \sin(10x)\), which shows that \(y\) depends on \(x\). To solve the problem, we need to find how these functions are connected by differentiating the composite function.Understanding a composite function is crucial, as it allows the application of the chain rule. The chain rule helps us break down and differentiate each part step-by-step.

Differentiation

Differentiation involves finding the derivative of a function. The derivative represents the rate of change of the function's value.
In our exercise, we need to find the derivative of \(z\) with respect to \(x\). The expression, \(z = (x + y)^5\), makes it clear that both \(x\) and \(y\) affect \(z\).
Differentiation is typically done step-by-step:

• First, differentiate \(z\) with respect to \(y\), treating \(y\) as a variable.
• Second, find the derivative of \(y\) with respect to \(x\).
• Then, use the chain rule to combine these derivatives.

This stepwise differentiation ensures we accurately determine how small changes in \(x\) affect \(z\).

Partial Derivatives

Partial derivatives are used when a function depends on more than one variable. They help us understand how the function changes with respect to one variable, while keeping others constant.
In our exercise, while \(z\) depends on both \(x\) and \(y\), we often hold one constant to differentiate with respect to the other.
For example:

• The partial derivative of \(z\) with respect to \(y\) is \(\frac{\partial z}{\partial y} = 5(x + y)^4\).
• The partial derivative of \(y\) with respect to \(x\) is \(\frac{\partial y}{\partial x} = 10 \cos(10x)\).

Using these partial derivatives, we apply the chain rule to find the total derivative \(\frac{dz}{dx}\).
By combining the steps, you get the overall change in \(z\) due to changes in \(x\).

Trigonometric Functions

Trigonometric functions are functions related to angles. They often come up in calculus problems, such as our exercise. In our case, we have the function \(y = \sin(10x)\), a trigonometric function depending on \(x\).
To differentiate \(y\) with respect to \(x\), we use our knowledge of trigonometric derivatives:

• The derivative of \(\sin(u)\) is \(\cos(u) * \frac{du}{dx}\).

Thus, given \(y = \sin(10x)\), we differentiate it as:

• \(\frac{dy}{dx} = 10 \cos(10x)\).

Combining this with the derivative of \(z\) with respect to \(y\), we apply the chain rule to find \(\frac{dz}{dx}\):

• \(\frac{dz}{dx} = 5(x + \sin(10x))^4 * 10 \cos(10x)\).

Understanding trigonometric functions and their derivatives is vital for solving such calculus problems.