Differentiating Implicitly

Use implicit differentiation to find dy/dx in Exercises 1–14.

x⁴ + sin y = x³y²

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Start by differentiating both sides of the equation with respect to x. Remember that y is a function of x, so when differentiating terms involving y, use the chain rule.

Differentiate the left side: The derivative of x⁴ with respect to x is 4x³. For sin(y), use the chain rule: the derivative is cos(y) * dy/dx.

Differentiate the right side: Use the product rule for x³y². The product rule states that d(uv)/dx = u'v + uv'. Here, u = x³ and v = y². Differentiate u to get 3x² and v to get 2y * dy/dx.

Substitute the derivatives back into the equation: 4x³ + cos(y) * dy/dx = 3x²y² + x³ * 2y * dy/dx.

Rearrange the equation to solve for dy/dx. Collect all terms involving dy/dx on one side and factor dy/dx out. Then, solve for dy/dx by isolating it on one side of the equation.

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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. It involves differentiating both sides of an equation with respect to a variable, often x, while treating other variables, like y, as implicit functions of x. This method is essential when dealing with equations where y cannot be easily isolated.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In implicit differentiation, the chain rule is crucial when differentiating terms involving y, as it accounts for the derivative of y with respect to x.

Trigonometric Derivatives

Trigonometric derivatives are the derivatives of trigonometric functions, such as sine, cosine, and tangent. For example, the derivative of sin(y) with respect to y is cos(y). When using implicit differentiation, it's important to apply these derivatives correctly, especially when differentiating terms like sin(y) with respect to x, which involves using the chain rule to account for dy/dx.

Differentiating ImplicitlyUse implicit differentiation to find dy/dx in Exercises 1–14.x⁴ + sin y = x³y²