If f(x) = (x)(4^x), what is the simplified second derivative?

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11-03-2021
Mathematics

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If f(x) = (x)(4^x), what is the simplified second derivative?

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ninjapuppy ninjapuppy · Answer 1 · 2021-03-11T04:25:10+01:00

First find f’(x) using the product rule.

The product rule shows how to find the derivative of two terms being multiplied together. The formula is: f(x) = xy, f’(x) =(x)(y’)+(y)(x’)

The derivative of a^x = a^x(lna)

The derivative of x(4^x) is (x)(4^x(ln4))+(4^x)
which can be written as 4^x(ln(4)(x) +1)

To take the f’’(x), use the product rule again.
4^x(ln(4)) + (ln(4)(x) + 1)(4^xln4). You can factor out 4^x(ln(4)) to get 4^xln(4)(ln(4)(x)+1+1))

1+1=2. The final simplified answer is (4^xln(4))(ln(4)(x)+2). I don’t think you can simplify anymore