Calculus Differentiation Techniques and Exercises

Vocabulary and core calculus differentiation methods covering problems 3.5.48, EXTRA 3.59, 3.9.85, 3.7.66, and 3.10.32 from the Pearson MyLab homework assignments.

Exercise 3.5.48 Derivative Problem

The problem requiring the evaluation of the derivative $\frac{dy}{dt}$ for the trigonometric function $y = \frac{8\tan(t)}{3 + 3\sec(t)}$, which evaluates to $\frac{dy}{dt} = \frac{8\sec(t)}{3(1 + \sec(t))} = \frac{8}{3(1 + \cos(t))}$.

Quotient Rule

The differentiation identity applied to evaluate derivatives of ratios like $y = \frac{8\tan(t)}{3 + 3\sec(t)}$, stating that $\frac{d}{dt}\left[\frac{u(t)}{v(t)}\right] = \frac{u'(t)v(t) - u(t)v'(t)}{[v(t)]^2}$.

Exercise EXTRA 3.59 Implicit Relation

The equation $y\sqrt{x + y} = 12$, which requires evaluating and simplifying $y'$ using implicit differentiation to obtain $y' = -\frac{y}{2x + 3y}$.

Implicit Differentiation

A technique applied in Exercise EXTRA 3.59 where both sides of an equation in $x$ and $y$ are differentiated with respect to $x$, treating $y$ as a differentiable function of $x$, and algebraically isolating $y'$.

Exercise 3.9.85 Differentiation Problem

The problem requiring the derivative $\frac{d}{dx}\left[\left(1 + \frac{3}{x}\right)^x\right]$, which evaluates via logarithmic differentiation to $\left(1 + \frac{3}{x}\right)^x \left[\ln\left(1 + \frac{3}{x}\right) - \frac{3}{x + 3}\right]$.

Logarithmic Differentiation

The method specified in Exercise 3.9.85 for differentiating variable-base and variable-exponent expressions $y = [f(x)]^{g(x)}$ by taking the natural logarithm of both sides, expanding using logarithm rules, and differentiating implicitly.

Exercise 3.7.66 Derivative Problem

The problem requiring the derivative $\frac{dy}{dx}$ of $y = \tan(15xe^x)$, which requires applying the chain rule combined with the product rule to yield $\frac{dy}{dx} = 15e^x(1 + x)\sec^2(15xe^x)$.

Chain Rule

The formula used to compute the derivative of a composite function $y = f(g(x))$, given by $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$, applied in functions such as $\tan(15xe^x)$ and $\ln(\tan(15t^3))$.

Exercise 3.10.32 Derivative Problem

The problem of evaluating the derivative $f'(t)$ of the logarithmic-trigonometric composite function $f(t) = \ln(\tan(15t^3))$, which evaluates to $f'(t) = \frac{45t^2\sec^2(15t^3)}{\tan(15t^3)}$.

Derivative of the Natural Logarithm

The calculus rule stating that $\frac{d}{du}[\ln(u)] = \frac{1}{u}$, which combines with the chain rule to give $\frac{d}{dt}[\ln(u(t))] = \frac{u'(t)}{u(t)}$, as used in Exercise 3.10.32.