Find dy/dx y=4xarcsin(x)
The problem provided is a calculus problem that involves differentiation. Specifically, you are being asked to find the first derivative with respect to x (denoted dy/dx) of the function y = 4x * arcsin(x). Here, y is defined as a product of the variable x and the inverse sine function of x, which is often denoted as arcsin(x) or sin^(-1)(x). The question requires you to apply the rules of differentiation, such as the product rule and the chain rule, to compute the derivative of this function with respect to x.
$y = 4 x \left(sin\right)^{-1} \left(\right. x \left.\right)$
Answer
Solution:
Step 1:
Apply the derivative operator to both sides of the given function: $\frac{d}{dx}(y) = \frac{d}{dx}(4x\arcsin(x))$.
Step 2:
The derivative of $y$ with respect to $x$ is denoted as $\frac{dy}{dx}$.
Step 3:
Take the derivative of the right-hand side of the equation.
Step 3.1:
Recognize that the constant $4$ does not change with $x$, so the derivative is $4\frac{d}{dx}[x\arcsin(x)]$.
Step 3.2:
Apply the Product Rule for differentiation, which is given by $\frac{d}{dx}[uv] = u\frac{dv}{dx} + v\frac{du}{dx}$, where $u = x$ and $v = \arcsin(x)$. Thus, we have $4(x\frac{d}{dx}[\arcsin(x)] + \arcsin(x)\frac{d}{dx}[x])$.
Step 3.3:
The derivative of $\arcsin(x)$ with respect to $x$ is $\frac{1}{\sqrt{1 - x^2}}$. This gives us $4(x\frac{1}{\sqrt{1 - x^2}} + \arcsin(x)\frac{d}{dx}[x])$.
Step 3.4:
Proceed with differentiation using the Power Rule.
Step 3.4.1:
Combine the terms $x$ and $\frac{1}{\sqrt{1 - x^2}}$ to get $4(\frac{x}{\sqrt{1 - x^2}} + \arcsin(x)\frac{d}{dx}[x])$.
Step 3.4.2:
Apply the Power Rule, which states that $\frac{d}{dx}[x^n] = nx^{n-1}$ where $n = 1$, to differentiate $x$. This yields $4(\frac{x}{\sqrt{1 - x^2}} + \arcsin(x) \cdot 1)$.
Step 3.4.3:
Multiply $\arcsin(x)$ by $1$ to obtain $4(\frac{x}{\sqrt{1 - x^2}} + \arcsin(x))$.
Step 3.5:
Simplify the expression.
Step 3.5.1:
Apply the distributive property to get $4\frac{x}{\sqrt{1 - x^2}} + 4\arcsin(x)$.
Step 3.5.2:
Combine the constant $4$ with $\frac{x}{\sqrt{1 - x^2}}$ to obtain $\frac{4x}{\sqrt{1 - x^2}} + 4\arcsin(x)$.
Step 4:
Express the equation by equating the left side to the simplified right side: $y = \frac{4x}{\sqrt{1 - x^2}} + 4\arcsin(x)$.
Step 5:
Substitute $\frac{dy}{dx}$ for $y$ to get the final derivative: $\frac{dy}{dx} = \frac{4x}{\sqrt{1 - x^2}} + 4\arcsin(x)$.
Knowledge Notes:
Derivative Operator: The notation $\frac{d}{dx}$ represents the derivative of a function with respect to the variable $x$.
Product Rule: A rule in calculus used to find the derivative of the product of two functions. It states that $\frac{d}{dx}[uv] = u\frac{dv}{dx} + v\frac{du}{dx}$.
Power Rule: A basic rule of differentiation that states if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
Inverse Trigonometric Functions: Functions like $\arcsin(x)$, $\arccos(x)$, and $\arctan(x)$ are the inverse functions of the sine, cosine, and tangent functions, respectively. The derivative of $\arcsin(x)$ with respect to $x$ is $\frac{1}{\sqrt{1 - x^2}}$.
Simplification: The process of rewriting an expression in a simpler or more compact form, often by combining like terms or using algebraic properties.
Distributive Property: A property of multiplication over addition or subtraction, stating that $a(b + c) = ab + ac$. This property is often used in simplifying expressions and solving equations.
