Differentiating Implicitly


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


x cos(2x + 3y) = y sin x

Verified step by step guidance

1. 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.
2. Differentiate the left side: For x cos(2x + 3y), use the product rule. The derivative of x is 1, and the derivative of cos(2x + 3y) is -sin(2x + 3y) multiplied by the derivative of the inside function, which is 2 + 3(dy/dx).
3. Differentiate the right side: For y sin x, use the product rule again. The derivative of y is dy/dx, and the derivative of sin x is cos x.
4. Set the derivatives from both sides equal to each other. This will give you an equation involving dy/dx.
5. Solve the resulting equation for dy/dx to find the derivative of y with respect to x.

Key Concepts

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 often applied when differentiating terms involving y, as y is considered a function of x.

Trigonometric Derivatives

Trigonometric derivatives are formulas used to find the derivatives of trigonometric functions such as sine, cosine, and tangent. For example, the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). In implicit differentiation problems involving trigonometric functions, these derivatives are crucial for correctly differentiating terms like x cos(2x + 3y) and y sin x.