Math 100 Final Logarithmic Differentiation, Implicit Differentiation, and derivatives of Inverse Trigonometric Functions

"What is the first step in Implicit Differentiation?"

"Differentiate both sides of the equation with respect to $x$

"What is the derivative of $y^3$ with respect to $x$ in Implicit Differentiation?"

"$3y^2 \cdot y'$ (or $3y^2 \frac{dy}{dx}$)."

"When should you use Logarithmic Differentiation?"

"1. When the function has the form $y = f(x)^{g(x)}$ (variable in base and exponent). 2. When differentiating complex products or quotients with many terms."

"What is the derivative of $\ln y$ with respect to $x$?"

"$\frac{1}{y} \cdot y'$"

"What is the derivative of $\arcsin(x)$?"

"$\frac{1}{\sqrt{1-x^2}}$"

"What is the derivative of $\arctan(x)$?"

"$\frac{1}{1+x^2}$"

"What is the derivative of $\arccos(x)$?"

"$-\frac{1}{\sqrt{1-x^2}}$"

"What relationship is used to derive the derivative of $y = \arcsin x$ if you forget the formula?"

"$\sin y = x$. Differentiate implicitly to get $(\cos y)y' = 1$

"When should you use Implicit Differentiation?"

"Use it when $x$ and $y$ are mixed together in an equation and it is difficult or impossible to isolate $y$ on one side (e.g.

"What is the core concept behind Implicit Differentiation?"

"You assume that $y$ is actually a function of $x$ (i.e.

"What is the general procedure for Implicit Differentiation?"

"1. Differentiate both sides with respect to $x$. 2. Collect all terms with $y'$ on one side. 3. Factor out $y'$. 4. Divide to solve for $y'$."

"When is Logarithmic Differentiation strictly necessary (not just helpful)?"

"It is necessary when differentiating functions where both the base and the exponent are variables

"What is the core concept behind Logarithmic Differentiation?"

"It uses the properties of logarithms (specifically $\ln(a^b) = b \ln a$ and $\ln(ab) = \ln a + \ln b$) to simplify complicated functions before differentiating them."

"What is the general procedure for Logarithmic Differentiation?"

"1. Take $\ln$ of both sides. 2. Use log laws to simplify the right side. 3. Differentiate implicitly (remembering LHS becomes $y'/y$). 4. Multiply both sides by $y$ (the original function) to isolate $y'$."