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Derivative of sin(x) * e^x: Product Rule Example
Published on October 13, 2024
Calculus
Learn how to find the derivative of the function g(x) = sin(x) * e^x using the product rule of differentiation. The solution reveals that the derivative is g'(x) = e^x (sin(x) + cos(x)).
Question
Calculate the derivative of g(x) = sin(x) * e^x
Answer
To find the derivative of g(x) = sin(x) * e^x, we’ll use the product rule of differentiation. Here’s how it works:
1. The Product Rule
The product rule states that the derivative of a product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function.
Mathematically:
If u(x) and v(x) are both differentiable functions, then the derivative of their product is:
d(u(x)v(x))/dx = u(x) * dv(x)/dx + v(x) * du(x)/dx
2. Applying the Rule to Our Problem
- Let u(x) = sin(x)
- Let v(x) = e^x
Now, let’s find the derivatives of u(x) and v(x):
- du(x)/dx = cos(x) (The derivative of sin(x) is cos(x))
- dv(x)/dx = e^x (The derivative of e^x is e^x)
3. Putting It All Together
Now we can apply the product rule:
d(g(x))/dx = u(x) * dv(x)/dx + v(x) * du(x)/dx
= sin(x) * e^x + e^x * cos(x)
= sin(x) * e^x + e^x * cos(x)
4. Simplifying
We can factor out e^x to simplify the expression:
d(g(x))/dx = e^x (sin(x) + cos(x))
Therefore, the derivative of g(x) = sin(x) * e^x is g’(x) = e^x (sin(x) + cos(x)).