Consider the curve x=e^y. Use implicit differentiation to verify that dy/dx = e^-y and then find d²y/dx² .

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Start by differentiating both sides of the equation x = e^y with respect to x. This requires using implicit differentiation since y is a function of x.

Differentiate the left side: The derivative of x with respect to x is 1.

Differentiate the right side: Use the chain rule. The derivative of e^y with respect to y is e^y, and then multiply by dy/dx because of the chain rule. So, the derivative is e^y * (dy/dx).

Set the derivatives equal: 1 = e^y * (dy/dx). Solve for dy/dx by dividing both sides by e^y, giving dy/dx = 1/e^y, which simplifies to e^-y.

To find d²y/dx², differentiate dy/dx = e^-y with respect to x again. Use the chain rule: the derivative of e^-y with respect to y is -e^-y, and multiply by dy/dx. Substitute dy/dx = e^-y into this expression to find d²y/dx².

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Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. Instead of solving for one variable in terms of the other, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule as necessary. This method is particularly useful for curves defined by equations like x = e^y, where y is not isolated.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. It states that if a function y is defined as a function of u, which in turn is a function of x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is essential in implicit differentiation, where we often need to differentiate y with respect to x while treating y as a function of x.

Second Derivative

The second derivative, denoted as d²y/dx², measures the rate of change of the first derivative (dy/dx) with respect to the independent variable. It provides information about the curvature of the function and can indicate concavity or points of inflection. To find the second derivative in the context of implicit differentiation, we differentiate the first derivative again, applying the chain rule and product rule as needed to account for any dependencies between the variables.