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

The given problem is asking for the derivative of the function y with respect to x, where y is expressed as the cosine of x squared. In other words, the question requires applying the chain rule of calculus to find the rate of change of the function y = cos(x^2) as x changes. The chain rule is a formula to compute the derivative of a composite function.

$y = c o s (x^{2})$

Answer

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Solution:

Step 1:

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

Step 2:

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

Step 3:

Compute the derivative of the right-hand side.

Step 3.1:

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

Step 3.1.1:

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

Step 3.1.2:

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

Step 3.1.3:

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

Step 3.2:

Employ the power rule for differentiation.

Step 3.2.1:

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

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: $- 2 x \sin (x^{2})$ .

Step 4:

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

Step 5:

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

Knowledge Notes:

Derivative: The derivative of a function measures how the function value changes as its input changes. The notation $\frac{d}{d x}$ is used to denote the derivative with respect to $x$ .
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 $f^{'} (g (x)) \cdot g^{'} (x)$ .
Power Rule: A basic rule for differentiation which states that if $y = x^{n}$ , then the derivative of $y$ with respect to $x$ is $\frac{d y}{d x} = n x^{n - 1}$ .
Trigonometric Functions: Functions like $\sin (x)$ and $\cos (x)$ have well-known derivatives, which are $- \sin (x)$ and $- \cos (x)$ respectively.
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.