Derivative of e^(4x)*sin(5x)

Answer

4⋅3253681254⋅x⋅sin(5⋅x)1196962444⋅x+5⋅3253681254⋅x⋅cos(5⋅x)1196962444⋅x\frac{4 \cdot 325368125^{4 \cdot x} \cdot \mathrm{sin}\left(5 \cdot x\right)}{119696244^{4 \cdot x}}+\frac{5 \cdot 325368125^{4 \cdot x} \cdot \mathrm{cos}\left(5 \cdot x\right)}{119696244^{4 \cdot x}}
4*e^(4*x)*sin(5*x)+5*e^(4*x)*cos(5*x)

Derivative of e^(4x)*sin(5x)

Rule Used
Product Rule
Steps
  1. Identify a product of two functions of x in e4⋅xe^{4 \cdot x}*sin⁡(5⋅x)\sin(5 \cdot x): use the product rule (uv)' = u'v + uv'. Each factor is differentiated once while the other is held as-is, then the two terms are added.
  2. Differentiate each factor and combine.
  3. Final answer: 4⋅e4⋅x4 \cdot e^{4 \cdot x}*sin⁡(5⋅x)\sin(5 \cdot x)+5⋅e4⋅x5 \cdot e^{4 \cdot x}*cos⁡(5⋅x)\cos(5 \cdot x)

The derivative of e^(4*x)*sin(5*x) is 4*e^(4*x)*sin(5*x)+5*e^(4*x)*cos(5*x).

Instant step-by-step solution for e^(4x)*sin(5x)

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Derivative of e^(4x)*sin(5x) = 4*e^(4*x)*sin(5*x)+5*e^(4*x)*cos(5*x) Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-e-minus-to-minus-the-minus-4xsin5x

How to find the derivative of e^(4x)*sin(5x)

To find the derivative of e^(4x)*sin(5x), we use standard differentiation rules. Our AI-powered calculator breaks down the steps and explains the logic.

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Frequently Asked Questions

What is the derivative of e^(4x)*sin(5x)?▼
The derivative of e^(4x)*sin(5x) is 4*e^(4*x)*sin(5*x)+5*e^(4*x)*cos(5*x).
How do you find the derivative of e^(4x)*sin(5x)?▼
Apply the standard differentiation rules — the key rule used here is the Product Rule. The step-by-step result is 4*e^(4*x)*sin(5*x)+5*e^(4*x)*cos(5*x).
Why is the derivative of e^(4x)*sin(5x) equal to 4*e^(4*x)*sin(5*x)+5*e^(4*x)*cos(5*x)?▼
Because differentiation measures how a function changes — applying the rules to e^(4x)*sin(5x) yields 4*e^(4*x)*sin(5*x)+5*e^(4*x)*cos(5*x). See the full step-by-step breakdown above.

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