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

Solution:

Step 1:

Apply the derivative operator to both sides of the equation: $\frac{d}{dx}y = \frac{d}{dx}\cos(x^2)$.

Step 2:

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

Step 3:

Compute the derivative of the right-hand side.

Step 3.1:

Invoke the chain rule for differentiation, which is given by $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$, with $f(x) = \cos(x)$ and $g(x) = x^2$.

Step 3.1.1:

Introduce a substitution $u = x^2$ and differentiate: $\frac{d}{du}[\cos(u)] \cdot \frac{d}{dx}[x^2]$.

Step 3.1.2:

The derivative of $\cos(u)$ with respect to $u$ is $-\sin(u)$: $-\sin(u) \cdot \frac{d}{dx}[x^2]$.

Step 3.1.3:

Substitute back $u = x^2$: $-\sin(x^2) \cdot \frac{d}{dx}[x^2]$.

Step 3.2:

Employ the power rule for differentiation.

Step 3.2.1:

Apply the power rule which states $\frac{d}{dx}[x^n] = nx^{n-1}$, where $n = 2$: $-\sin(x^2) \cdot (2x)$.

Step 3.2.2:

Simplify the resulting expression.

Step 3.2.2.1:

Combine the constant factors: $-2 \cdot \sin(x^2) \cdot x$.

Step 3.2.2.2:

Rearrange the terms to form the final expression: $-2x \sin(x^2)$.

Step 4:

Construct the final equation by equating the left-hand side with the right-hand side: $\frac{dy}{dx} = -2x \sin(x^2)$.

Step 5:

Substitute $\frac{dy}{dx}$ for $y$ to complete the differentiation: $\frac{dy}{dx} = -2x \sin(x^2)$.

Knowledge Notes:

1. Derivative: The derivative of a function measures how the function value changes as its input changes. The notation $\frac{d}{dx}$ is used to denote the derivative with respect to $x$.

2. Chain Rule: A fundamental rule in calculus for finding the derivative of a composite function. It states that if $y = f(g(x))$, then the derivative of $y$ with respect to $x$ is $f'(g(x)) \cdot g'(x)$.

3. Power Rule: A basic rule for differentiation which states that if $y = x^n$, then the derivative of $y$ with respect to $x$ is $\frac{dy}{dx} = nx^{n-1}$.

4. Trigonometric Functions: Functions like $\sin(x)$ and $\cos(x)$ have well-known derivatives, which are $-\sin(x)$ and $-\cos(x)$ respectively.

5. Simplification: After applying the chain rule and power rule, it is often necessary to simplify the expression by combining like terms or constants to achieve the final derivative form.