Given:
[tex]F(x)=f(g(x))[/tex]
where, [tex]f(-1)=5,f'(-1)=3,f'(5)=3,g(5)=-1,g'(5)=8[/tex].
To find:
The value of [tex]F'(5)[/tex].
Solution:
We have,
[tex]F(x)=f(g(x))[/tex]
Differentiate with respect to x.
[tex]F'(x)=\dfrac{d}{dx}f(g(x))[/tex]
Using chain rule, we get
[tex]F'(x)=f'(g(x))g'(x)[/tex]
Now, put x=5.
[tex]F'(5)=f'(g(5))g'(5)[/tex]
[tex]F'(5)=f'(-1)\times 8[/tex] [tex][\because g(5)=-1,g'(5)=8][/tex]
[tex]F'(5)=3\times 8[/tex] [tex][\because f'(-1)=3][/tex]
[tex]F'(5)=24[/tex]
Therefore, the value of [tex]F'(5)[/tex] is 24.
