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

The question asks for the derivative of the function y with respect to x, where y is given as the product of x squared and the sine of 2x. To find this derivative, you would need to apply rules of differentiation such as the product rule and the chain rule.

$y = x^{2} s i n (2 x)$

Answer

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Solution:

Step 1:

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

Step 2:

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

Step 3:

Apply differentiation to the right-hand side.

Step 3.1:

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

Step 3.2:

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

Step 3.2.1:

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

Step 3.2.2:

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

Step 3.2.3:

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

Step 3.3:

Proceed with differentiation.

Step 3.3.1:

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

Step 3.3.2:

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

Step 3.3.3:

Simplify the terms.

Step 3.3.3.1:

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

Step 3.3.3.2:

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

Step 3.3.4:

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

Step 3.3.5:

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

Step 4:

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

Step 5:

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

Knowledge Notes:

Product Rule: When differentiating a product of two functions, $u (x)$ and $v (x)$ , the derivative is given by $u^{'} v + u v^{'}$ .
Chain Rule: This rule is used when differentiating a composite function, $f (g (x))$ . The derivative is $f^{'} (g (x)) g^{'} (x)$ .
Power Rule: For any real number $n$ , the derivative of $x^{n}$ with respect to $x$ is $n x^{n - 1}$ .
Derivative of Trigonometric Functions: The derivative of $\sin (x)$ is $\cos (x)$ , and the derivative of $\cos (x)$ is $- \sin (x)$ .
Constants: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function.
Simplification: After applying differentiation rules, it's important to simplify the expression to make the result clearer and easier to understand.