If y = e^{3x} \sin(2x), what is \frac{dy}{dx}? — JAMB UTME

If $y = e^{3x} \sin(2x)$ , what is $\frac{dy}{dx}$ ?

e^{3x}(3\sin 2x + 2\cos 2x)

Using the product rule

\frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}

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

gives

e^{3x} \cdot 2\cos(2x) + \sin(2x) \cdot 3e^{3x}

. Factoring out

e^{3x}

results in

e^{3x}(3\sin 2x + 2\cos 2x)

Identify the function components for the product rule

Let

u = e^{3x}

and

v = \sin(2x)

The function

y

is a product of two differentiable functions

u(x)

and

v(x)

Differentiate each component using the chain rule

\frac{du}{dx} = 3e^{3x}

and

\frac{dv}{dx} = 2\cos(2x)

Applying the chain rule gives

\frac{d}{dx}(e^{3x}) = 3e^{3x}

and

\frac{d}{dx}(\sin 2x) = 2\cos 2x

Apply the product rule formula

\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}

\frac{dy}{dx} = e^{3x}(2\cos 2x) + \sin(2x)(3e^{3x}) = e^{3x}(3\sin 2x + 2\cos 2x)

Combining the products and factoring out the common exponential factor

e^{3x}

yields the final derivative.

Differentiation of Transcendental Functions using Product Rule and Chain Rule

Estimated Time:1m 30s

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