Analytical Applications of Differentiation

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 $\frac{dy}{dx}$.

$(0, 5)$

$(-3, -4)$

$(0, -5)$

$(3, 4)$

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

The curve passes through ($a, b$).

The partial derivatives with respect to both variables exist and are continuous near ($a, b$).

There exists no horizontal or vertical tangent line at ($a, b$).

The total derivative with respect to either variable is zero at ($a, b$).

The effect cannot be determined without additional information about $x$, $y$, or both.

It increases the slope because it effectively raises the height of level curves.

There is no effect on slopes since only individual values of $x$ and $y$ are influenced, not their rates of change.

It decreases the slope due to reducing pressure on both variables to increase simultaneously.

$2x$

$-\frac{x}{y}$

$-2y$

$2y$

Evaluating definite integrals

Finding the limit of a function

Solving differential equations

Finding areas under curves

t

y

z

x

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.

Integrate both sides with respect to x.

Take the partial derivative with respect to x.

Perform implicit differentiation to find $\frac{dy}{dx}$.

Solve for y in terms of x and then take the derivative.