Question

second order derivative of x^4.e^5x

Answer 1

2nd order derivative= [25x⁴+40x³+12x²]e^(5x)

Answer 2

Second order derivative of $x^{4} e^{5x}$ is $e^{5x} (25x^{2} +40x^{3} +12x^{2} )$.

Given,

The expression is $x^{4} e^{5x}$.

To find,

Second order derivative of $x^{4} e^{5x}$.

Solution,

Let,

y= $x^{4} e^{5x}$

Differentiating both sides with respect to x,

We get,

$\frac{dy}{dx} = x^{4} . 5e^{5x} + e^{5x} . 4x^{3}$

It is the first-order derivative.

Again, differentiating both sides with respect to x.

We get,

$\frac{d^2y}{dx^2} =x^4.(25e^{5x} )+e^{5x}.4x^{3} +e^{5x}.12x^{2} +4x^{3} .5e^{5x}$

$\frac{d^2y}{dx^2} = e^{5x} (25x^{2} +40x^{3} +12x^{2} )$

Second order derivative of $x^{4} e^{5x}$ is $e^{5x} (25x^{2} +40x^{3} +12x^{2} )$.

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