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.

$y = 4 x {(s i n)}^{- 1} (x)$

Answer

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Solution:

Step 1:

Apply the derivative operator to both sides of the given function: $\frac{d}{d x} (y) = \frac{d}{d x} (4 x \arcsin (x))$ .

Step 2:

The derivative of $y$ with respect to $x$ is denoted as $\frac{d y}{d x}$ .

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 $4 \frac{d}{d x} [x \arcsin (x)]$ .

Step 3.2:

Apply the Product Rule for differentiation, which is given by $\frac{d}{d x} [u v] = u \frac{d v}{d x} + v \frac{d u}{d x}$ , where $u = x$ and $v = \arcsin (x)$ . Thus, we have $4 (x \frac{d}{d x} [\arcsin (x)] + \arcsin (x) \frac{d}{d x} [x])$ .

Step 3.3:

The derivative of $\arcsin (x)$ with respect to $x$ is $\frac{1}{\sqrt{1 - x^{2}}}$ . This gives us $4 (x \frac{1}{\sqrt{1 - x^{2}}} + \arcsin (x) \frac{d}{d x} [x])$ .

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 $4 (\frac{x}{\sqrt{1 - x^{2}}} + \arcsin (x) \frac{d}{d x} [x])$ .

Step 3.4.2:

Apply the Power Rule, which states that $\frac{d}{d x} [x^{n}] = n x^{n - 1}$ 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 $\frac{4 x}{\sqrt{1 - x^{2}}} + 4 \arcsin (x)$ .

Step 4:

Express the equation by equating the left side to the simplified right side: $y = \frac{4 x}{\sqrt{1 - x^{2}}} + 4 \arcsin (x)$ .

Step 5:

Substitute $\frac{d y}{d x}$ for $y$ to get the final derivative: $\frac{d y}{d x} = \frac{4 x}{\sqrt{1 - x^{2}}} + 4 \arcsin (x)$ .

Knowledge Notes:

Derivative Operator: The notation $\frac{d}{d x}$ 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 $\frac{d}{d x} [u v] = u \frac{d v}{d x} + v \frac{d u}{d x}$ .
Power Rule: A basic rule of differentiation that states if $f (x) = x^{n}$ , then $f^{'} (x) = n x^{n - 1}$ .
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 $a (b + c) = a b + a c$ . This property is often used in simplifying expressions and solving equations.