Second derivatives Find d²y/dx²for the following functions.
y = x cos x²

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First, identify the function y = x cos(x²). We need to find the second derivative d²y/dx².

Start by finding the first derivative dy/dx. Use the product rule, which states that if you have a function y = u*v, then dy/dx = u'v + uv'. Here, u = x and v = cos(x²).

Calculate the derivative of u = x, which is u' = 1. Then, calculate the derivative of v = cos(x²) using the chain rule. The chain rule states that if you have a composite function f(g(x)), then the derivative is f'(g(x)) * g'(x). Here, f(x) = cos(x) and g(x) = x², so v' = -sin(x²) * 2x.

Substitute u', v, u, and v' into the product rule formula: dy/dx = 1 * cos(x²) + x * (-sin(x²) * 2x). Simplify this expression to get the first derivative.

Now, find the second derivative d²y/dx² by differentiating the expression obtained for dy/dx. Apply the product rule and chain rule again as needed, and simplify the resulting expression to obtain d²y/dx².

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Second Derivative

The second derivative of a function measures the rate of change of the first derivative, providing information about the curvature of the function's graph. It is denoted as d²y/dx² and is essential for understanding the acceleration of the function's values. In practical terms, it helps identify concavity and points of inflection.

Product Rule

The product rule is a fundamental differentiation technique used when finding the derivative of the product of two functions. It states that if u(x) and v(x) are functions, then the derivative of their product is given by d(uv)/dx = u'v + uv'. This rule is crucial for differentiating functions like y = x cos(x²), where two functions are multiplied.

Chain Rule

The chain rule is a method for differentiating composite functions, which are functions within functions. It states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). This rule is particularly important when dealing with functions like cos(x²), where the inner function x² requires differentiation to find the overall derivative.

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