Find the inverse of the function f(x) = x + 2/3x - 4 Let y = g(x) be the inverse

Question

Find the inverse of the function f(x) = x + 2/3x - 4 Let y = g(x) be the inverse function of f(x) = Squareroot 1 + x. Calculate g'(5). Let y = g(x) be the inverse function of f(x) = x^3 + 3x - 1. Calculate g'(3). Compute the derivative of f(x) = x^2 tan^-1 x. Find the equation of the line tangent to the graph of f(x) = sin^-1(3x) at x = 1/6 Use the formula g'(x) = 1/f'(g(x))' where y = is the inverse function for y = f(x) to compute g'(2) when f(x) = x^3 + x + 2. Let f(x) = x^3/2 - 2/x^3 on the interval 0

Explanation / Answer

1.Sol:- Solve (x+2)/(3x-4) =y for x

then x = (-4y-2)/(-3y+1)

replacing x in place of y we get

f-1(x) = (-4x-2)/(-3x+1)

2.Sol:- Solve (x+1) =y for x

then x = y2-1

replacing x in place of y we get

f-1(x) = g(x) = x2-1

Now g-1(x) = 2x

then g-1(5) = 10

3.Sol:-

Solve x3+3x-1 =y for x

then x = {3y+(y2+2y+5) +1}/32 - {32/3y+(y2+2y+5) +1}

replacing x in place of y we get

f-1(x) = g(x) = {3x+(x2+2x+5) +1}/32 - {32/3x+(x2+2x+5) +1}

4.Sol:- Apply the product rule

(f.g)' = f'g +f.g'

d/dx (x2) .tan-1x + d/dx (tan-1x) .x2

we get

x2/1+x2 + 2x(tan-1x)

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