Let for all real .

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

Let

f(x)=e^{2x}\sin 3x

for all real

x

. (a) Compute

f'(x)

. (b) Compute

f''(x)

and express your answer entirely in terms of

e^{2x}\sin 3x

and

e^{2x}\cos 3x

. (c) Derive a closed‐form expression for

f^{(n)}(x)

valid for every integer

n\ge0

. Your formula should involve

e^{2x}

multiplied by a linear combination of

\sin 3x

and

\cos 3x

whose coefficients depend on

n

. Show all steps that establish the pattern. (d) Evaluate

f^{(12)}(0)

Related theory

Differentiation Rules
Higher-Order Derivatives
Advanced Differentiation
Second order derivatives
Continuity and Differentiability

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