How do I solve these Calculus problems?

Question

I know its kind of a lot but I'm really struggling and I need help, thank you!

1. Find the equation of the tangent line to the inverse at the given point
2. F(x)= 7x+sin(2x) @(0,0)
3. F(x)= x^3+8x+cos(3x) @(1,0)
4. F(x)= 10x+(arc tanx)^2 @(0,0)
5. F(x)= 7x^3+(ln(x))^3 @(7,1)
6. A function F and its derivative take on the values shown in the table. If G is the inverse of F, find derivative G' (-6)
7. Table:
8. x=2 then f(x)=6 then f'(x)=1/3
9. x=2 then f(x)=8 then f'(x)=3/2
10. Let y=x^3+x+x-2, and let g be the inverse function. Find g'(0)

Answer 1

1

a) F(x)=7x+sin(2x), at (0, 0)

Derivative of the F(x): F'(x)= 7+2cos(2x) --> F'(0)=7

Derivative of the inverse function is

(F^-1)'(F(x))=1/(F'(x))= 1/(7+2cos(2x))

Hence (F^-1)'(F(0))=(F^-1)'(0)=1/9. The slope is 1/9.

Equation of the tangent has the form of y-y₀=a(x-x₀), (x_0, y₀)=(0, 0)

So the tangent is: y=(1/9)x or y=x/9

Other examples are solved in similar way.