Find dy/dx y=xcos(x)

The question provided is asking for the first derivative of the function y with respect to x, where y is defined as the product of x and the cosine of x. The derivative dy/dx represents the rate at which y changes with a small change in x. To find this derivative, one would use differentiation rules applicable to products of functions, specifically the product rule, which is used when taking the derivative of a product of two functions.

$y = x c o s (x)$

Answer

Expert–verified

Solution:

Step 1:

Apply the derivative operator to both sides of the given equation: $\frac{d}{d x} (y) = \frac{d}{d x} (x \cos (x))$ .

Step 2:

The derivative of $y$ in terms of $x$ is denoted as $\frac{d y}{d x}$ .

Step 3:

Proceed to differentiate the right-hand side of the equation.

Step 3.1:

Utilize the Product Rule for differentiation, which is given by $\frac{d}{d x} [u (x) v (x)] = u (x) \frac{d}{d x} [v (x)] + v (x) \frac{d}{d x} [u (x)]$ , where $u (x) = x$ and $v (x) = \cos (x)$ . Thus, we get $x \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x]$ .

Step 3.2:

Compute the derivative of $\cos (x)$ with respect to $x$ , which is $- \sin (x)$ , leading to $x (- \sin (x)) + \cos (x) \frac{d}{d x} [x]$ .

Step 3.3:

Apply the Power Rule for differentiation.

Step 3.3.1:

The Power Rule states that $\frac{d}{d x} [x^{n}] = n x^{n - 1}$ , where $n = 1$ . This simplifies to $x (- \sin (x)) + \cos (x) \cdot 1$ .

Step 3.3.2:

Simplify the resulting expression.

Step 3.3.2.1:

Multiply $\cos (x)$ by $1$ to obtain $x (- \sin (x)) + \cos (x)$ .

Step 3.3.2.2:

Rearrange the terms to get the final derivative: $- x \sin (x) + \cos (x)$ .

Step 4:

Combine the results to form the complete derivative equation: $\frac{d y}{d x} = - x \sin (x) + \cos (x)$ .

Step 5:

Substitute $\frac{d y}{d x}$ for $y$ to express the final answer: $\frac{d y}{d x} = - x \sin (x) + \cos (x)$ .

Knowledge Notes:

Derivative Operator: The derivative operator $\frac{d}{d x}$ is used to denote the rate of change of a function with respect to the variable $x$ .
Product Rule: The Product Rule is a fundamental rule in calculus used to differentiate products of two functions. It states that the derivative of a product $u (x) v (x)$ is $u^{'} (x) v (x) + u (x) v^{'} (x)$ .
Power Rule: The Power Rule is used to differentiate functions of the form $x^{n}$ , where $n$ is any real number. The rule states that the derivative of $x^{n}$ is $n x^{n - 1}$ .
Trigonometric Functions: The derivatives of basic trigonometric functions are essential in calculus. For instance, the derivative of $\sin (x)$ is $\cos (x)$ , and the derivative of $\cos (x)$ is $- \sin (x)$ .
Simplification: After applying differentiation rules, expressions are often simplified to make them more concise and easier to interpret.
Notation: It is important to use the correct notation when differentiating. For example, $\frac{d y}{d x}$ represents the derivative of $y$ with respect to $x$ .