Derivative Questions

Question

Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number.

 

Let f be an increasing function with f(0)=3. The derivative of f is given by f′(x)=cos(πx)+x4+6.

a) Find f′′(−3).

b) Write an equation for the line tangent to the graph of y=(f(x))^2 at x=0.

c) Let g be the function defined by g(x)=f(√2x^2+7). Find g′(3).

d) Let h be the inverse function of f. Find h′(3).

Answer 1

a)

f’’(x) = -π*cos(πx)+4x^3

f’’(-3) = -π*cos(-3π)+4(-3)^3 = -108

b)

dy/dx = 2*f(x)*f’(x)

At x=0, dy/dx = 2*f(0)*f’(0) = 2*3*[cos(0)+0^4+6] = 6*7=42.

This is the slope of the tangent line.

When x=0, y=(f(0))^2=9.

So equation of the tangent line: y-9=42(x-0), i.e. y=42x+9

c)

g(x) = f(√2x^2+7)

g’(x) = f’(√2x^2+7)*(2√2x) = (2√2x)[-π*sin(π(√2x^2+7))+4*(√2x^2+7)^3]

g’(3) = 260617.62… (plug in 3 into above)

d)

f(h(x))=x

d/dx[f(h(x))] = d/dx[x]

f’(h(x))*h’(x)=1

h’(x) = 1/f’(h(x))

Since f(0)=3, h(3)=0.

So h’(3) = 1/f’(0) = 1/[cos(0*pi)+0^4+6] = 1/7