Part $3$: Thinking $[16$ marks$]$

$23.$ Differentiate the following.

b$)y=5-e^x$

$[1$ mark$]$

a$)y=5^{2x^2},[1$ mark$]$

If $y=2sinx-cosx,$ find $y^{\text{'}\text{'}}$ at $x=\frac{}{3}.$

a$.\frac{1+\sqrt[\phantom{2}]{3}}{2}$

c$.\frac{1-2\sqrt[\phantom{2}]{3}}{2}$

b$.\frac{1-\sqrt[\phantom{2}]{3}}{2}$

d$.\frac{1+2\sqrt[\phantom{2}]{3}}{2}$

If $y=\frac{sin3x}{9},$ find $y^{\text{'}\text{'}}$ at $x=\frac{}{6}.$

a$.1$

c$.0$

b$.-1$

d$.\sqrt[\phantom{2}]{2}$

Find the slope of the tangent to $y=1-2si{n}^{2}x$ at the point where $x=\frac{}{4}.$

a$.1$

c$.2$

b$.-1$

d$.-2$

If $f(x)=sin3x,$ find $f^{\text{'}\text{'}}(\frac{}{6}).$

a$.3$

c$.9$

b$.-3$

d$.-9$

If $f(x)=co{s}^2(nx)-si{n}^2(nx),$ find $f^{\text{'}\text{'}\text{'}}(\frac{}{4n}).$

a$.-8n^3$

c$.8n^3$

b$.-8$

d$.-4n^2$

Find $f^{\text{'}}(x)$ if $f(x)=2e^x.$

a$.2e^x$

c$.e^{x}ln2$

b$.e^x$

d$.2e$

Find the equation of the tangent to $y=e^{sinx}$ at the point $(0,1).$

a$.y=x+$

c$.y=ex+1$

b$.y=-x-1$

d$.y=x+1$

If $f(x)=2x^2{e}^x,$ find $f^{\text{'}}(1).$

a$.2e$

c$.0$

b$.6e$

d$.4e$

Find the equation of the tangent to $y=3e^x+3ex$ that is parallel to $y=6ex+5.$

a$.6ex-y=0$

c$.6ex-y=6e$

b$.6ex+y=0$

d$.6ex+y=6e$