Q2 Iffx 6 XINX find f6 Use implicit differentiation to find

Q2:

Iff(x) = 6 XIN(X), find f\'(6). Use implicit differentiation to find the slope of the tangent line to the curve xy3 + xy = 18 at the point (9,1). xy3 + xy = 18 at the point (9, 1).

Solution

1)Let y = f(x)
=>y = 6x^ln(x)

=>ln(y) = ln(6x^ln(x))

=>ln(y) = ln(6) + ln(x^ln(x))

=>ln(y) = ln(6) + ln(x)ln(x)

Differentiating (1) w.r.t to x, we get

=>(1/y)*(dy/dx) = 0 + (1/x)*ln(x) + ln(x)*(1/x) = (2/x)*ln(x)

=>dy/dx) = (2y/x)*ln(x)

=>f\'(x) = (2f(x)/x)*ln(x)

=>f\'(6) = (2f(6)/6)*ln(6) = 88.832

2)xy³ + xy = 18

Differentiating w.r.t. x we get,

=>y³ + 3xy²*(dy/dx) + y + x*(dy/dx) = 0

=>dy/dx*(3xy²+x) = -(y+y³)

=>dy/dx = -(y+y³)/(3xy²+x)

Therefore dy/dx at (9,1) = -1/18