In Exercises 9 and 10, use implicit differentiation to find dy/dx.
9. y^e^x = x^y + 1

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Start with the given equation: \(y^{e^{x}} = x^{y} + 1\).

Differentiate both sides of the equation with respect to \(x\) using implicit differentiation. Remember that \(y\) is a function of \(x\), so when differentiating terms involving \(y\), apply the chain rule.

For the left side, \(y^{e^{x}}\), use logarithmic differentiation: take the natural logarithm of both sides or apply the formula for differentiating \(u^{v}\) where both \(u\) and \(v\) depend on \(x\). The derivative is \(\frac{d}{dx} y^{e^{x}} = y^{e^{x}} \left( e^{x} \ln(y) + e^{x} \frac{1}{y} \frac{dy}{dx} \right)\).

For the right side, differentiate \(x^{y}\) using implicit differentiation: \(\frac{d}{dx} x^{y} = x^{y} \left( \frac{y}{x} + \ln(x) \frac{dy}{dx} \right)\), and the derivative of the constant \(1\) is \(0\).

After differentiating both sides, collect all terms involving \(\frac{dy}{dx}\) on one side and factor it out. Then solve for \(\frac{dy}{dx}\) to express it explicitly in terms of \(x\), \(y\), and known functions.

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Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function when y is defined implicitly rather than explicitly in terms of x. It involves differentiating both sides of the equation with respect to x, treating y as a function of x, and applying the chain rule when differentiating terms involving y.

Chain Rule

The chain rule is a fundamental differentiation rule used when differentiating composite functions. In implicit differentiation, it allows us to differentiate terms involving y by multiplying the derivative of the outer function by dy/dx, since y depends on x.

Logarithmic Differentiation

Logarithmic differentiation is useful when differentiating expressions where variables appear as both bases and exponents, such as y^e^x or x^y. Taking the natural logarithm of both sides simplifies the differentiation process by converting products and powers into sums and products, making implicit differentiation more manageable.

In Exercises 9 and 10, use implicit differentiation to find dy/dx.9. y^e^x = x^y + 1