13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
sin y = 5x⁴−5; (1, π)

Verified step by step guidance

1. Start by differentiating both sides of the equation with respect to x. The equation is \( \sin y = 5x^4 - 5 \).
2. For the left side, differentiate \( \sin y \) with respect to y, which gives \( \cos y \), and then multiply by \( \frac{dy}{dx} \) due to the chain rule. This results in \( \cos y \cdot \frac{dy}{dx} \).
3. For the right side, differentiate \( 5x^4 - 5 \) with respect to x. The derivative of \( 5x^4 \) is \( 20x^3 \), and the derivative of \(-5\) is 0. So, the right side becomes \( 20x^3 \).
4. Set the derivatives equal to each other: \( \cos y \cdot \frac{dy}{dx} = 20x^3 \).
5. Solve for \( \frac{dy}{dx} \) by dividing both sides by \( \cos y \), resulting in \( \frac{dy}{dx} = \frac{20x^3}{\cos y} \).

Key Concepts

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 y in terms of x, we differentiate both sides of the equation with respect to x, treating y as a function of x. This method is particularly useful for equations that are difficult or impossible to rearrange.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. When using implicit differentiation, the chain rule is applied to terms involving y, resulting in the derivative dy/dx. This means that when differentiating a function of y, we multiply by dy/dx to account for the dependence of y on x.

Evaluating at a Point

After finding the derivative dy/dx using implicit differentiation, we often need to evaluate it at a specific point, such as (1, π) in this case. This involves substituting the x and y values into the derived expression to find the slope of the tangent line at that point. This step is crucial for understanding the behavior of the function at specific coordinates.