Find derivatives for the functions. Assume a, b, c, and k are constants. f(t)=3t^{2}-4t+1

Use the given graph to estimate the value of each derivative.(Round all answers to one decimal place.)Graph uploaded below.
(a) f ' (0)1
(b) f ' (1)2
(c) f ' (2)3
(d) f ' (3)4
(e) f ' (4)5
(f) f ' (5)6

What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders? Give examples.

Compute the inverse function of the following polynomial on $[0,1]$?
$f(x)=\alpha x^3-2\alpha x^2+(\alpha +1)x$
(where $\alpha$ is within $]0,3[$)

Find the derivatives of the functions. ${\displaystyle f\left(x\right)=[\ln \left(e^{x^2}\right)]-\ln \left[\left(e^{x^2}\right)\right]}$

Limit $\underset{x\to \mathrm{\infty }}{lim}(\sin \sqrt{1+x}-\sin \sqrt{x})$

Calculating $\underset{x\to 0}{lim}\frac{\sec (x)-1}{x^2\sec (x)}$
The first thing to do, as I was taught, was to rewrite this in terms of sine and cosine. Since $\sec (x)=\frac{1}{\cos (x)}$ we have
$\frac{\frac{1}{\cos (x)}-1}{x^2\frac{1}{\cos (x)}}$
And that is
$\frac{\frac{1-\cos (x)}{\cos (x)}}{\frac{x^2}{\cos (x)}}$
Then
$\frac{(1-\cos (x))\cdot \cos (x)}{\cos (x)\cdot x^2}$
And
$\frac{(1-\cos (x))}{x^2}$
What was wrong with my procedure?

Can the functions be added, subtracted, multiplied, and divided except where the denominator is zero to produce new functions?