How would I differentiate a function such as f(x)=x^3(e^(2x))?
Here, f(x)=x3e2x is a function consisting of two functions multiplied together, so we need to use the product rule. The product rule is as follows: where f(x)=u(x)v(x), f'(x)=u(x)v'(x)+u'(x)v(x). The first step involves identifying the two functions that are multiplied together, and representing them by u and v. So, let u(x)=x3 and v(x)=e2x. Now, we must find u'(x) and v'(x). u'(x)=3x2 (from Core 1: multiply by the power, then subtract 1 from the power) and v'(x)=2e2x (from using the chain rule). Then, substitute u(x), v(x), u'(x) and v'(x) into our product rule formula, giving f'(x)=x3 . 2e2x + 3x2 .e2x If you wish to simplify this, you can do so by taking out a common factor of x2e2x from each term: f'(x)=x2e2x(2x+3)
LB
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