If f(x)= 3sin(x) + 4(x)^x

Question

If f(x)=3sin(x) + 4(x)^x, find f'(3)

Answer 1

f(x) = 3 sin x + 4 x^x

f’(x) = (3 sin x + 4 x^x)’
       = (3 sin x)’ + (4 x^x)’                                term by term derivative

                                                                    (sin x)' = cos x
       = 3 cos x + 4 [(e^{ln x})^x]’                            x = e^{ln x} [1]
       = 3 cos x + 4 (e^{x ln x})’                             (a^b)^c = a^bc [2]
       = 3 cox x + 4 (e^{x ln x})(x ln x)’                  (e^u)’ = e^u u’ (chain rule)
       = 3 cos x + 4 (e^{ln x})^x (x ln x)’                  according to [2]
       = 3 cos x + 4 x^x (x ln x)’                         according to [1]
       = 3 cos x + 4 x^x [(1) ln x + x (1/x)]         (fg)’ = f’g + fg’ (product rule)
                                                                    (ln x)’ = 1/x
       = 3 cos x + 4 x^x (ln x + 1)

F’(3) = 3 cos 3 + (4)(3³)(ln 3 + 1)
       = 3 cos 3 + 108 (1 + ln 3)
       ≅ 223.68

Answer 2

Let's denote these quantities with variables.

 

D = diameter = 16 cm

V = volume of sphere = (4/3)π(D/2)³

d/dt = rate

 

dD/dt = 0.4  

 

We need to find dV/dt.

 

Given these variables, we can establish a formula:

 

dV/dt = (dV/dD)*(dD/dt)

 

In order to use this formula, we need to find dV/dD.  This means find the derivative of the volume in terms of the diameter.

 

V = (4/3)π(D/2)³

 

 simplifies to

 

V = (1/6)πD³

 

dV/dD = (1/2)πD²

 

 

Substitute  D=16  into this derivative.

 

dV/dD = (1/2)π(16)²

 

dV/dD = 402.12

 

 

Going back to our main formula,

 

dV/dt = (dV/dD)*(dD/dt)

 

dV/dt = (402.12)(0.4)

 

dV/dt = 160.85 cm³ / min