Pre calculus Compute ddx fgx where fx squareroot x and gx

Compute d/dx f(g(x)), where f(x) = squareroot x and g(x) = x^2 +2x+7 Write each expression in the form 2^kx or 3^kx for a suitable constant k. 8^x 4(squareroot 9)^x (1/27)^x

Solution

f(x)=x^1.2

g(x)=x^2 +2x+7

f(g(x))=f(x^2 + 2x+7)= (x^2 +2x+7)^1/2

therefore d/dxf(g(x))= d/dx(x^2+2x+7)^1/2= (1/2)(x^2 +2x +7)(2x+2)

                                                             =(x+1)/(x^2 +2x+7)

4. a. 8^x

8=2^3

therefore 8^x=2^(3x)

b. (fourth root 9)^x

9=3^2

therefore

(fourth root 9)^x=(3^2/4)^x= 3^x/2

c.(1/27)^x

27=3^3

therefore 1/27=3^-3

(1/27)^x = 3^-3x