Question

find d²y/dx² if y = e⁵x​

Answer 1

y=tan−1x

x=tany

Differentiating w.r.t.y, we get

dydx=sec2y

Diff both sides w.r.t x, 

dx2d2y=dxd(cos2y)

dx2d2y=−2cosysinydxdy

dx2d2y=−2cosysiny×cos2y

dx2d2y=−2sinycos3y

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Answer 2

Given :-

y = ${e}^{5x}$

Differentiating the function w.r.t 'x'

dy/dx = 5${e}^{5x}$ -----(1)

As we know that, when a function is in the form of y = e^(nx)

After differentiating we get dy/dx = n•x^(nx)

Again differentiating equation (1) w.r.t x we get :-

d²y/dx² = 25${e}^{5x}$

There will be same method like before :-

Double differentiation of a function y = e^(nx)

d²y/dx² = n²•e^(nx)