Composite, Implicit, and Inverse Functions

$72x^2 + 12x - 10$

$12x^2 + 6x - 5$

$72x + 12$

$72$

2

0

2x

x^4

Any real number k excluding zero.

Only when k equals to zero.

For any integer value of k not equal to one.

Only when k equals to one over a to power n.

The specific coordinates where $h(x)$ crosses axes

The concavity or inflection points based on sign changes

The slope or directionality at that point

The maximum or minimum values around that point

$\frac{d^{4}y}{dx^{4}}=12e^{2t}$

$\frac{d^{4}y}{dx^{4}}=48e^{2t}$

$\frac{d^{4}y}{dx^{4}}=24e^{2t}$

$\frac{d^{4}y}{dx^{4}}=36e^{2t}$

-$\\cos( \\frac{\\pi}{4} )$

$\\sqrt{2}$

$-\\sqrt{2}$

-$\\sin( \\frac{\\pi}{4} )$

$25e^{5k}$

$125e^{5k}$

$625e^{5k}$

$20e^{5k}$

$4\cos(x) + 24\sin(2x)$

$-4(6\sin(2x) + \cos(x))$

$-4\sin(x) - 24\cos(2x)$

$-4\sin(x) + 12\cos(2x)$

$24x^2 - 16x + 2$

$12x^3 - 16x + 1$

$12x^3 - 8x + 2$

$36x^2 - 16$

Integrating $f''(x)$ twice and then differentiating thrice.

Calculating the Taylor series expansion about $x=0$ and finding the third coefficient.

Applying the chain rule three times and simplifying as you go.

Using the definition of a derivative to calculate each order successively.