Find dy/dx y=x^2sin(2x)

Solution:

Step 1:

Take the derivative of both sides with respect to $x$: $\frac{d}{dx}(y) = \frac{d}{dx}(x^2 \sin(2x))$

Step 2:

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

Step 3:

Apply differentiation to the right-hand side.

Step 3.1:

Utilize the Product Rule: $\frac{d}{dx}[uv] = u\frac{dv}{dx} + v\frac{du}{dx}$, where $u = x^2$ and $v = \sin(2x)$, to get $x^2 \frac{d}{dx}[\sin(2x)] + \sin(2x) \frac{d}{dx}[x^2]$.

Step 3.2:

Employ the Chain Rule: $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$, with $f(x) = \sin(x)$ and $g(x) = 2x$.

Step 3.2.1:

Introduce $u = 2x$ to apply the Chain Rule: $x^2 \left(\frac{d}{du}[\sin(u)] \frac{d}{dx}[2x]\right) + \sin(2x) \frac{d}{dx}[x^2]$.

Step 3.2.2:

Compute the derivative of $\sin(u)$ with respect to $u$: $x^2 \left(\cos(u) \frac{d}{dx}[2x]\right) + \sin(2x) \frac{d}{dx}[x^2]$.

Step 3.2.3:

Substitute back $2x$ for $u$: $x^2 \left(\cos(2x) \frac{d}{dx}[2x]\right) + \sin(2x) \frac{d}{dx}[x^2]$.

Step 3.3:

Proceed with differentiation.

Step 3.3.1:

Recognize that $2$ is a constant, thus the derivative of $2x$ is $2$: $x^2 \left(\cos(2x) (2)\right) + \sin(2x) \frac{d}{dx}[x^2]$.

Step 3.3.2:

Apply the Power Rule: $\frac{d}{dx}[x^n] = nx^{n-1}$, where $n = 1$: $x^2 \left(\cos(2x) (2 \cdot 1)\right) + \sin(2x) \frac{d}{dx}[x^2]$.

Step 3.3.3:

Simplify the terms.

Step 3.3.3.1:

Multiply $2$ by $1$: $x^2 (2 \cdot \cos(2x)) + \sin(2x) \frac{d}{dx}[x^2]$.

Step 3.3.3.2:

Reposition $2$ in front of $\cos(2x)$: $x^2 (2 \cdot \cos(2x)) + \sin(2x) \frac{d}{dx}[x^2]$.

Step 3.3.4:

Apply the Power Rule again for $n = 2$: $x^2 (2 \cos(2x)) + \sin(2x) (2x)$.

Step 3.3.5:

Rearrange the expression: $2x^2 \cos(2x) + 2x \sin(2x)$.

Step 4:

Combine the terms to form the final expression: $y = 2x^2 \cos(2x) + 2x \sin(2x)$.

Step 5:

Replace $y$ with $\frac{dy}{dx}$ to get the derivative: $\frac{dy}{dx} = 2x^2 \cos(2x) + 2x \sin(2x)$.

Knowledge Notes:

1. Product Rule: When differentiating a product of two functions, $u(x)$ and $v(x)$, the derivative is given by $u'v + uv'$.

2. Chain Rule: This rule is used when differentiating a composite function, $f(g(x))$. The derivative is $f'(g(x))g'(x)$.

3. Power Rule: For any real number $n$, the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.

4. Derivative of Trigonometric Functions: The derivative of $\sin(x)$ is $\cos(x)$, and the derivative of $\cos(x)$ is $-\sin(x)$.

5. Constants: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.

6. Simplification: After applying differentiation rules, it's important to simplify the expression to make the result clearer and easier to understand.