Differentiate the following functions and simplify. (Might need to use logarithmic properties if necessary

Question

PLEASE DO ALL 3 QUESTIONS

1) f(x) = tan (ln(4x-9))

2) f(x) = log₅(x⁴√(2x² + 9x))

3) y = e^{(8x + π)^4} ln(2x-1)

Answer 1

To differentiate these functions, multiple applications of the chain rule and the product rule suffice. For instance, we can write y = e^{(8x + π)^4} ln(2x-1) as y = f(g(h(x)))*l((m(x)), where f(x) = e^x , g(x) = x⁴, h(x) = 8x+π, l(x) = ln(x), and m(x) is 2x-1. Then the chain rule plus the product rule says the derivative is

y' = f'(g(h(x))*g'(h(x))*h'(x)*l(m(x)) + f(g(h(x))*l'(m(x))*m'(x).

This is 32(8x+π)³e^{(8x + π)^4} ln(2x-1)+e^{(8x + π)^4}(2/(2x-1)).

If you need a tool to do them, Wolfram Alpha will differentiate them easily enough.