Find dr/dθ in Exercises 15–18.

r – 2√θ = (3/2)θ²/³ + (4/3)θ³/⁴

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Start by identifying the given equation: r - 2√θ = (3/2)θ²/³ + (4/3)θ³/⁴. We need to find dr/dθ, which is the derivative of r with respect to θ.

Rearrange the equation to express r explicitly in terms of θ: r = 2√θ + (3/2)θ²/³ + (4/3)θ³/⁴.

Differentiate each term of the equation with respect to θ. For the term 2√θ, use the derivative of θ^(1/2), which is (1/2)θ^(-1/2).

For the term (3/2)θ²/³, apply the power rule: the derivative is (3/2) * (2/3)θ^(2/3 - 1). Simplify this expression.

For the term (4/3)θ³/⁴, again use the power rule: the derivative is (4/3) * (3/4)θ^(3/4 - 1). Simplify this expression. Combine all the derivatives to find dr/dθ.

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주요 개념

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Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function when it is not explicitly solved for one variable in terms of another. In this problem, r is given implicitly in terms of θ, so we differentiate both sides of the equation with respect to θ, treating r as a function of θ, and then solve for dr/dθ.

Power Rule for Differentiation

The power rule is a basic rule in calculus used to differentiate functions of the form x^n, where n is a real number. The derivative of x^n is n*x^(n-1). In this problem, terms like θ^(2/3) and θ^(3/4) require applying the power rule to find their derivatives with respect to θ.

Chain Rule

The chain rule is used to differentiate composite functions. It states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). In this context, when differentiating terms involving θ raised to a power, the chain rule helps in handling the differentiation of nested functions, especially when θ is under a radical or fractional exponent.

Find dr/dθ in Exercises 15–18.r – 2√θ = (3/2)θ²/³ + (4/3)θ³/⁴