Question

Easy Quotient/Product Rule Question

f(x) = (x^4)(cos x)^-1
I know the answer is x^4 tan x + 4x^3 sec x
But when I did the problem again today, I kept coming up with a different answer. Could someone solve with steps?

Answer 1

using the product rule

u = x^4 du/dx = 4x^3

v = (cos(x))^-1 dv/du = -1 (cos(x))^-2 * -sin(x)

so

\[\frac{dy}{dx} = u \times \frac{dv}{dx} + v \times \frac{du}{dx}\]

which gives

\[\frac{dy}{dx} = x^4 \times \sin(x)(\cos(x)^{-2} + (\cos(x))^{-1} \times 4x^3\]

which can be simplified to

\[\frac{dy}{dx} = \frac{x^4 \sin(x)}{(\cos(x))^2} + \frac{4x^3}{\cos(x)}\]

Answer 2

Thanks so much!