$g\left(x\right)=ax^2+bx+c$ such that $g\left(0\right)=1, g'\left(1\right)=g\left(1\right)$ and $g'\left(-1\right)=g\left(-1\right)$.
Find the values of $a, b$ and $c$.
$a=$
$b=$
$c=$
For $f\left(x\right)=\left(x+1\right)\left(x^2+1\right)\left(x^4+1\right)\left(x^8+1\right)\left(x^{16}+1\right)$, find the value of $f'\left(2\right)$.
$f'\left(2\right)=$
$\left(2\right)$$+$
If $f'\left(x\right)+f\left(x\right)=x^3+2x^2+5x+4$, find $f\left(x\right)$.
$f\left(x\right)=$
$f\left(x\right)=ax^2+bx-2$ where $a$ and $b$ are constants.
Find the values of $a$ and $b$ if $f\left(f'\left(x\right)\right)=f'\left(f\left(x\right)\right)$
$a=$
$b=$
If $f”\left(x\right)=af\left(x\right)+bf’\left(x\right)$ where $f\left(x\right)=e^{2x}\sin x$, find the values of $a$ and $b$.
If $f(x)=x^3+ax^2+bx+c$ and $\left(x-2\right)f'(x)=3f(x)$, find the values of $a, b$ and $c$.
$a=$
$b=$
$c=$
Find $a>1$ such that $a^x=x$ has one unique solution.
$a=$
For $\displaystyle f(x)=\frac{x^{n+1}}{\left(n+1\right)^2}\Bigl\{\left(n+1\right) \ln (x)-1 \Bigl\}$, find $f'(x)$ in terms of $x$ and $n$.
Find the derivative of:
*If your answer includes logarithms, use $\ln$
a)$x^x \left(x>0\right)$
b)$x^{\ln x} \left(x>0\right)$
If $f”\left(x\right)=af\left(x\right)+bf’\left(x\right)$ where $f\left(x\right)=e^{2x}\sin x$, find the values of $a$ and $b$.