Differentiate y=e^(x)*sin(x) with respect to x

To find the derivative of the function \( y = e^{x} \sin(x) \) with respect to \( x \), we will apply the product rule of differentiation. The product rule states that if you have a function that is the product of two differentiable functions, say \( u(x) \) and \( v(x) \), then the derivative of their product is given by \( (uv)’ = u’v + uv’ \). In this case, we can identify \( u(x) = e^{x} \) and \( v(x) = \sin(x) \).

First, we need to compute the derivatives of \( u(x) \) and \( v(x) \). The derivative of \( u(x) = e^{x} \) is \( u'(x) = e^{x} \), since the exponential function is its own derivative. For \( v(x) = \sin(x) \), the derivative is \( v'(x) = \cos(x) \). Now, substituting these derivatives into the product rule formula, we have \( y’ = u’v + uv’ = e^{x} \sin(x) + e^{x} \cos(x) \).

Combining these results, we can express the derivative of the original function in a more compact form. Thus, the derivative of \( y = e^{x} \sin(x) \) with respect to \( x \) is given by \( y’ = e^{x} (\sin(x) + \cos(x)) \). This expression succinctly captures the rate of change of the function \( y \) in relation to \( x \), incorporating both the sine and cosine components multiplied by the exponential factor.