Nth Derivative Of `e^{2x} Sin3x.Sin2x` | Leibnitz Theorem Of Nth Derivatives Examples

To Find N^th Derivative of e^2x sin3x sin2x

In this tutorial, we learn how to find the n^th derivative of expressions involving e^ax · sin(bx) · sin(cx) using identities and standard nth derivative formulas.

Solution

Let
u = e^2x
v = sin3x · sin2x

Using Trigonometric Identity

sinA · sinB = ½ [ cos(A − B) − cos(A + B) ]

A = 3x
B = 2x

∴ A − B = x
∴ A + B = 5x

Rewrite the Function

y = e^2x · ½ [ cos x − cos 5x ]
= ½ [ e^2x cos x − e^2x cos 5x ]

Standard Formula

If y = e^Ax cos(Bx + C), then

dⁿy/dxⁿ = (A² + B²)^n/2 e^Ax cos(Bx + C + n tan⁻¹(B/A))

N^th Derivative of e^2x cos x

= ½ · 5^n/2 e^2x cos(x + n tan⁻¹(1/2))

N^th Derivative of e^2x cos 5x

= ½ · 29^n/2 e^2x cos(5x + n tan⁻¹(5/2))

Final Answer

½ · 5^n/2 e^2x cos(x + n tan⁻¹(1/2)) − ½ · 29^n/2 e^2x cos(5x + n tan⁻¹(5/2))

Recommended Mathematics Articles

• Nth Derivative Using Successive Differentiation
• Nth Derivative Using Leibnitz Rule
• Nth Derivative of xⁿyⁿ

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