$1$ Find the derivative $(\frac{dy}{dx})$ of the following functions with respect to $x$ :

$($a$)y=x^5$

$($b$)y=e^{4x}$

$($c$)y=lnx$

$($d$)y=sin(2x)$

$2$ Evaluate the following integrals:

$($a$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}15xdx$

$($b$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}3x^2-4dx$

$($c$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{3}{x^4}dx$

$($d$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}5x^2+2cos(x)dx$

$3$ By using the substitution $t=3x+1,$ find ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(12(3x+1{)}^3)dx.$

$4($a$)$ Given the function of a curve C is : $y=x^3-9x^2+24x+5$

$($i$)$ Find $\frac{dy}{dx}$ and $\frac{d^{2}y}{d{x}^2}$; and

$($ii$)$ Find the local maximum and minimum value of $y$ by using your results obtained in part $($i$)$;$.$

$($b$)($i$)$ Determine the equation of the tangent of C at the point $(0,5)$; and

$($ii$)$ the equation of the normal of C at the point $(0,5).$