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

검증된 단계별 안내

Start by differentiating both sides of the equation with respect to x. The equation is \( \sin y = 5x^4 - 5 \).

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 \( \cos y \cdot \frac{dy}{dx} \).

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 \( 20x^3 \).

Set the derivatives equal to each other: \( \cos y \cdot \frac{dy}{dx} = 20x^3 \).

Solve for \( \frac{dy}{dx} \) by dividing both sides by \( \cos y \), resulting in \( \frac{dy}{dx} = \frac{20x^3}{\cos y} \).

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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 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.

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