Derivative Calculations

In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).

y = √u, u = sin x

Verified step by step guidance

First, identify the functions involved: y = √u and u = sin(x). We need to find dy/dx using the chain rule.

The chain rule states that dy/dx = f'(u) * g'(x), where f(u) = √u and g(x) = sin(x).

Calculate f'(u): Since y = √u, f(u) = u^(1/2). The derivative f'(u) with respect to u is (1/2)u^(-1/2).

Calculate g'(x): Since u = sin(x), the derivative g'(x) with respect to x is cos(x).

Combine the derivatives using the chain rule: dy/dx = f'(u) * g'(x) = (1/2)u^(-1/2) * cos(x). Substitute u = sin(x) into the expression to get dy/dx = (1/2)(sin(x))^(-1/2) * cos(x).

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Key Concepts

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

Chain Rule

The chain rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if you have a function y = f(u) and u = g(x), then the derivative dy/dx is found by multiplying the derivative of f with respect to u, f'(u), by the derivative of u with respect to x, g'(x). This allows us to differentiate complex functions by breaking them down into simpler parts.

Derivative of Square Root Function

To differentiate a square root function, y = √u, we use the power rule. The square root can be expressed as u^(1/2), and its derivative is (1/2)u^(-1/2) or 1/(2√u). This derivative is crucial when applying the chain rule to composite functions involving square roots, as it provides the rate of change of the square root function with respect to its variable.

Derivative of Sine Function

The derivative of the sine function, u = sin x, is cos x. This is a basic derivative that is essential when applying the chain rule to functions involving trigonometric components. Understanding this derivative allows us to determine how the sine function changes with respect to x, which is necessary for calculating the overall derivative of composite functions like y = √u where u = sin x.

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