Find dy/dx y=9xarcsin(x)

Solution:

Step:1

Apply the differentiation operator to both sides of the given function: $\frac{d}{dx}(y) = \frac{d}{dx}(9x\arcsin(x))$

Step:2

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

Step:3

Proceed to differentiate the expression on the right-hand side.

Step:3.1

Recognize that the constant $9$ can be factored out of the differentiation process: $9\frac{d}{dx}(x\arcsin(x))$

Step:3.2

Apply the product rule for differentiation: $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$, where $u = x$ and $v = \arcsin(x)$: $9\left(x\frac{d}{dx}(\arcsin(x)) + \arcsin(x)\frac{d}{dx}(x)\right)$

Step:3.3

Find the derivative of $\arcsin(x)$ with respect to $x$: $9\left(x\frac{1}{\sqrt{1 - x^2}} + \arcsin(x)\frac{d}{dx}(x)\right)$

Step:3.4

Utilize the power rule for differentiation.

Step:3.4.1

Combine terms involving $x$ and $\frac{1}{\sqrt{1 - x^2}}$: $9\left(\frac{x}{\sqrt{1 - x^2}} + \arcsin(x)\frac{d}{dx}(x)\right)$

Step:3.4.2

Differentiate $x$ using the power rule: $\frac{d}{dx}(x^n) = nx^{n-1}$, where $n = 1$: $9\left(\frac{x}{\sqrt{1 - x^2}} + \arcsin(x) \cdot 1\right)$

Step:3.4.3

Simplify the multiplication of $\arcsin(x)$ by $1$: $9\left(\frac{x}{\sqrt{1 - x^2}} + \arcsin(x)\right)$

Step:3.5

Simplify the expression.

Step:3.5.1

Apply the distributive property to combine like terms: $9\frac{x}{\sqrt{1 - x^2}} + 9\arcsin(x)$

Step:3.5.2

Factor out the constant $9$: $\frac{9x}{\sqrt{1 - x^2}} + 9\arcsin(x)$

Step:4

Express the derivative of $y$ with respect to $x$: $y' = \frac{9x}{\sqrt{1 - x^2}} + 9\arcsin(x)$

Step:5

Replace $y$ with $\frac{dy}{dx}$ in the final result: $\frac{dy}{dx} = \frac{9x}{\sqrt{1 - x^2}} + 9\arcsin(x)$

Knowledge Notes:

The problem involves finding the derivative of a function that is a product of a constant, a variable, and an inverse trigonometric function. The solution requires knowledge of several calculus concepts:

1. Derivative Notation: $\frac{d}{dx}$ represents the differentiation operator with respect to $x$.

2. Product Rule: A rule for differentiating products of two functions, given by $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$.

3. Derivative of Inverse Trigonometric Functions: Specifically, the derivative of $\arcsin(x)$ is $\frac{1}{\sqrt{1 - x^2}}$.

4. Power Rule: A basic rule for differentiation, stating that the derivative of $x^n$ is $nx^{n-1}$.

5. Simplification: Combining like terms and applying algebraic rules to express the derivative in its simplest form.

These concepts are fundamental in differential calculus and are often used in combination to solve more complex differentiation problems.