Question

Question #e3127

Answer 1

`color(blue)(d/dxf(x)=2tan(x).sec^2(x)+ln(a).a^x)`

Explanation:

`f(x)=tan^2(x)+a^x`

`f(x)=sin^2(x)/cos^2(x)+a^x`

`f(x)=(1-cos^2(x))/cos^2(x)+a^x`

`f(x)=1/cos^2(x)-1+a^x`

`f(x)=1/((1+cos2x)/2)-1+a^x`

`f(x)=2/(1+cos2x)-1+a^x`

`f(x)=2/(1+cos2x)-1+e^(xln(a))`

`d/dxf(x)=(-2(-2sin(2x)))/(1+cos2x)^2+ln(a).e^(xln(a))=`

`=(4sin(2x))/(1+cos2x)^2+ln(a).a^x`

`=(8sin(x)cos(x))/[2cos^2(x)]^2+ln(a).a^x`

`=(8sin(x)cos(x))/[4cos^4(x)]+ln(a).a^x`

`=2(sin(x))/[cos^3(x)]+ln(a).a^x`

`=2tan(x)/[cos^2(x)]+ln(a).a^x`

`color(blue)(=2tan(x).sec^2(x)+ln(a).a^x`

Answer 2

`2tanxsec^2x+a^xlna`

Explanation:

`"differentiate "tan^2x"using the "color(blue)"chain rule"`

`" given "y=f(g(x))" then"`

`dy/dx=f'(g(x))xxg'(x)`

`rArrd/dx(tan^2x)=2tanx xxd/dx(tanx)=2tanxsec^2x`

`•color(white)(x)d/dx(a^x)=a^xlnalarrcolor(blue)"standard derivative"`

`rArrd/dx(tan^2x+a^x)`

`=2tanxsec^2x+a^xlna`