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.
$y = x cos \left(\right. x \left.\right)$
Answer
Solution:
Step 1:
Apply the derivative operator to both sides of the given equation: $\frac{d}{dx}(y) = \frac{d}{dx}(x\cos(x))$.
Step 2:
The derivative of $y$ in terms of $x$ is denoted as $\frac{dy}{dx}$.
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 $\frac{d}{dx}[u(x)v(x)] = u(x)\frac{d}{dx}[v(x)] + v(x)\frac{d}{dx}[u(x)]$, where $u(x) = x$ and $v(x) = \cos(x)$. Thus, we get $x\frac{d}{dx}[\cos(x)] + \cos(x)\frac{d}{dx}[x]$.
Step 3.2:
Compute the derivative of $\cos(x)$ with respect to $x$, which is $-\sin(x)$, leading to $x(-\sin(x)) + \cos(x)\frac{d}{dx}[x]$.
Step 3.3:
Apply the Power Rule for differentiation.
Step 3.3.1:
The Power Rule states that $\frac{d}{dx}[x^n] = nx^{n-1}$, 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: $\frac{dy}{dx} = -x\sin(x) + \cos(x)$.
Step 5:
Substitute $\frac{dy}{dx}$ for $y$ to express the final answer: $\frac{dy}{dx} = -x\sin(x) + \cos(x)$.
Knowledge Notes:
Derivative Operator: The derivative operator $\frac{d}{dx}$ 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 $nx^{n-1}$.
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, $\frac{dy}{dx}$ represents the derivative of $y$ with respect to $x$.
