Composite, Implicit, and Inverse Functions

Step 2

Step 1

Step 3

Implicit differentiation does not involve factoring out y'

Solving for $y$ first and then differentiating, as this method can be more complex due to the transcendental nature of the equation.

Graphical analysis, since it provides a visual approach but does not easily yield an exact numerical derivative.

Implicit differentiation, because it allows the calculation of $\frac{dy}{dx}$ directly within the given relationship between $x$ and $y$.

Numerical approximation methods such as Newton's Method, while useful in some cases, are less direct than implicit differentiation for this problem.

$-1$

$e$

$-\dfrac{e-1}{e+1}$

$0$

$-(e - e)$

$-\frac{d(e^{-1})}{dx}$

$-\frac{e^{-1}}{e} - {} + {}$

$-\frac{e^{-1}}{(e^{}} - {} + {}) - e \frac{d(e^{-1})}{dx}$

Whenever there are exponential or trigonometric relations between the variables because they require the application of the chain rule whether solved outright or implicitly.

In instances where every term contains y raised to different powers, thus making the task of isolation arduous, tedious, and nonrewarding.

Incorrect. These are the opposite scenarios which mixes both variables in equal measure without a clear advantage for either methodology.

When the function describing y in terms of x has simple algebraic structures that make explicit separation a straightforward and efficient option.

Using Implicit Differentiation Is More Accurate Because The Complicated Nature Of This Logarithm-Based Relation Leads To Increased Potential Errors When Attempting To Isolate One Variable Before Differentiating

Explicit Differentiation After Solving For Y May Seem Simplistic But Often Results In Less Accuracy Due To Its Unwieldiness With Complex Logarithmic Expression

Utilizing Graphical Analysis Provides A Visual Understanding That Can Potentially Minimize Computational Error, Though Not As Quantitative As Algebraic Methods

Applying Numerical Approximation Techniques Like Series Expansion Could Reduce Accuracy Compared To Direct Algebraic Approaches, Especially Near Points Where Convergence Is Slow

$\frac{-ye^{yx^2}}{y \cos(xy)-e^{yx^2} \cdot x \sin(xy)}$

$\frac{- \sin(yy)e^{xx^2}}{x \cos(xx)+yy}$

$\frac{e^{yx^2}}{y \cos(xy)-e^{yx^2}y}$

$\frac{-ye^{yx^2} \cdot x \sin(xy)}{x \cos(xy)+y \sin(xy)}$

$\frac{dy}{dx} = \frac{1}{2y}$

$\frac{dy}{dx} = \frac{2}{4x - 3}$

$\frac{dy}{dx} = \frac{1}{4x - 3}$

$\frac{dy}{dx} = \frac{4}{2y}$

$(y' + x) + (y^2)' = \frac{dy}{dx}$

$(y + xy') + (2yy') = 0$

$(xy)' + (y^2)' = \frac{d}{dx}$

$(1+y') - 2y'y = 0$

$\frac{15(\ln(x))^5}{(\ln(x))+20}$

$\frac{20(\ln(x))^4}{x (\ln(x))^5+25}$

$\frac{15(\ln(x))^4}{(\ln(x))^5+20 (\ln(x))+20}$

$\frac{12(\ln(x))^5}{x (\ln(x))+25}$