DIFFERENTIATION AND DERIVATIVE

What is the d/dx x^tanx?

y=x^tanx

taking log on both side

⇒logy=log(x^tanx)

or

⇒logy=tanx.logx

⇒differentiate both side

⇒(1/y)y’=tanx.(1/x)+logx.sec^2x

⇒y’={tanx/x+(logx.sec^2x)}*y
⇒y’={tanx/x+(logx.sec^2x)}*x^tanx

What is dydx     of y=xxx
xxxxxx


This function can be written as

y=x^y

take log on both sides

logy=logx^y

^logy=ylogx

^{diffentiate on both sides}

^{1/y(dy/dx)=y/x+logx(dy/dx)}

^{dy/dx(1/y-logx)=y/x}

^dy/dx=(y/x)/(1/y-logx)
What is the value of the derivative of log (base a) x?

y=log_ax

y=logx/loga

dy/dx=(1/x)/loga

dy/dx=log_ae/x

What is the differentiation of (sinx)cscx?

What's the derivative of ln(sinx cos x)?

y=ln(sinxcosx) 

using chain rule

dy/dx=1/(sinxcosx) d/dx(sinxcosx)

dy/dx=1/(sinxcosx)((sinx(-sinx)+cosx(cosx))

dy/dx=(cos²x-sin²x)/(sinxcosx) 

dy/dx=cos2x/(sinxcosx)

What's the differentiation of log(ax)?

y=ln(ax)

Using chain rule

dy/dx=(1/ax)d/dx(ax)

dy/dx=(1/ax)a

dy/dx=1/x

or 

y=ln(ax)

y=lna+lnx

dy/dx=1/x

How can we differentiate y=2^x?

y=2^x

^{take log on both sides}

^logy=xlog2

^{diff both sides}

^{dy/dx(1/y)=log2}

^dy/dx=ylog2

^dy/dx=2^xlog2

What is the 2019th derivative of sin(x) +cos(x)?

y=sinx+cosx

first derivative y’=cosx-sinx

second derivative y’’=-sinx-cosx

third derivative y’“=-cosx+sinx

fourth derivative =sinx+cosx

hence the 2016th will be same as fourth derivtive

and 2019th will be same as third derivative i.e.=sinx-cosx

Find the value of d2x/dx2 at the point defined by the given value of t.

x = 5 sin t, y = 5 cos t, t = 3π/4

x = 5 sin t, y = 5 cos t, t = 3π/4

D²Y/D²X=(D²Y/D²t/(D²x/D²t)

dy/dx=-5sint

D²Y/D²t=-5cost    and

dx/dt=5cost

D²x/D²t=-5sint

D²Y/D²X=(D²Y/D²t/(D²x/D²t)=(-5cost /-5sint)

=cot(t)

=cot(3Π/4)

=-1