Analytical Applications of Differentiation

Anywhere that either sine or cosine functions have discontinuities or infinite slopes.

At every integer multiple of $\pi$ because sine and cosine values repeat periodically.

Only when either side of equation equates zero because sine inverse does not exist there.

Wherever $\tan(xy)$ would be undefined since it reflects slopes directly related here.

Uniquely determined by applying implicit differentiation twice and evaluating at $(\pi/2,1)$.

Dependent on the value of $x$, which cannot be evaluated at a specific point.

Does not exist because $y$ is not explicitly defined as a function of $x$.

Equivalent to zero since $\sin(x)$ has an inflection point at $\pi/2$.

$y'(t)'$

$y'(x)'$

$y''(t)$

$y''(x)$

The magnitude $|\frac{dy}{dx}|$ increases.

The derivative stays consistent since it depends on the relation between c and xy, rather than an isolated change in c.

The denominator decreases due to decreasing y-values, leading to a higher rate of change in x.

The numerator increases due to the logarithmic term, slightly raising $\frac{dy}{dx}$.

The area under the curve

The rate of change of the y-coordinate

The concavity of the curve

The rate of change of the slope

$2x$

$-\frac{x}{y}$

$-2y$

$2y$

Evaluating definite integrals

Finding the limit of a function

Solving differential equations

Finding areas under curves

t

y

z

x

Explicit derivatives only

Implicit derivatives only

None of the above

Both implicit and explicit derivatives

Treat them as constants

Differentiate them with respect to x

Replace them with y

Replace them with x