A table of values for $ f, g, f' , $ and $ g' $ is given.
(a) If $ h(x) = f(g(x)), $ find $ h'(1). $
(b) If $ H(x) = g(g(x)), $ find $ H(1). $
for party This problem we want to find aged prime of one and we know that h of x is f of G of X. So let's use the chain rule to find h prime at X h. Prime of X will be the derivative of the outside f prime of Jew vex times the derivative of the inside G prime of X. Now let's substitute one in their h prime of one will be f prime of G of one times g prime of one. Now we can refer to the table g of one is to and g prime of one is six so we can substitute those numbers in so h prime of one is f prime of to times six. Now we need to find f prime of to looking at the table f prime of two is five so we have five times six So the answer is 30 And for part B, we're finding h prime of one for capital H and capital H is g of f of X. So the derivative of capital age according to the chain rule will be the derivative of the outside G prime of f of X times a derivative of the inside F prime of X. So now let's evaluate that at one h prime of one will be G prime of F one times f prime of one. Now let's use the table to find f of one and effort one is three and f prime of one f prime of one is for. So let's substitute those numbers in and we have aged. Prime of one equals G prime of three times four. Now let's find you a prime of three. J Prime of three is nine. So let's substitute that in and we have nine times for so our answer is 36.