DERIVATIVE OF A FUNCTION OF FUNCTION

Let y = f(u) and u = g(x).

Then the derivative of y with respect to x is

Find the derivative of each function with respect to x.

Problem 1 :

y = (x2 + 5x + 6)3

Solution :

y = (x2 + 5x + 6)3

Let u = x2 + 5x + 6.

Then, we have

y = u3 and u = x2 + 5x + 6

Problem 2 :

Solution :

Let u = x3 + 5.

Then, we have

y = eu and u = x3 + 5

Problem 3 :

Solution :

Let u = x2 - 11x + 24.

Then, we have

Problem 4 :

Solution :

Let u = ln x.

Then, we have

Problem 5 :

y = ln (1 + x2)

Solution :

Let u = 1 + x2.

Then, we have

y = ln u and u = 1 + x2

Problem 6 :

y = esin x

Solution :

Let u = sin x.

Then, we have

y = eu and u = sin x

Problem 7 :

y = ln (tan x)

Solution :

Let u = tan x.

Then, we have

y = ln u and u = tan x

Problem 8 :

y = tan (cos x)

Solution :

Let u = cos x.

Then, we have

y = tan u and u = cos x

Problem 9 :

y = cos (sin 2x)

Solution :

Let u = sin 2x and v = 2x.

Then, we have

y = cos u, = sin v and v = 2x

Problem 10 :

y = sin (sin x2)

Solution :

Let u = sin x2 and v = x2.

Then, we have

y = sin u, = sin v and v = x2

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