USING CHAIN RULE TO DIFFERENTIATE

Question 1 :

Differentiate F(x)  =  (x³ + 4x)⁷

Solution :

F(x)  =  (x³ + 4x)⁷

Let u = x³ + 4x

Differentiate the function "u" with respect to x, we get

du/dx  =  3x² + 4(1)

  =  3x² + 4

F(x) = u⁷

Differentiate the function "y" with respect to x, we get

F'(x)  =  7u⁶ (du/dx)

F'(x)  =  7(x³ + 4x)⁶ (3x² + 4)

Question 2 :

Differentiate h(x)  =  (t - (1/t))^3/2

Solution :

h(x)  =  (t - (1/t))^3/2

Let u = t - (1/t)

Differentiate the function "u" with respect to x, we get

du/dx  =  1 + (1/t²)

h(x) = u^3/2

Differentiate the function "y" with respect to x, we get

h'(x)  =  (3/2)u^1/2 (du/dx)

h'(x)  =  (3/2)(t - (1/t))^1/2(1 + (1/t²))

Question 3 :

Differentiate f(t)  =  ∛(1 + tan t)

Solution :

f(t)  =  ∛(1 + tan t)

Let u = 1 + tan t

Differentiate the function "u" with respect to x, we get

du/dx  =  0 + sec²t

=  sec²t

f(t) = u^1/3

Differentiate the function "y" with respect to x, we get

f'(t)  =  (1/3)u^-2/3 (du/dx)

f'(t)  =  (1/3)(1 + tan t)^-2/3 (sec²t)

f'(t)  =  (1/3)sec²t(1 + tan t)^-2/3

Question 4 :

Differentiate y  =  cos (a³ + x³)

Solution :

y  =  cos (a³ + x³)

Let u = a³ + x³

Differentiate the function "u" with respect to x, we get

du/dx  =  0 + 3x²

=  3x²

y = cos u

Differentiate the function "y" with respect to x, we get

dy/dx  =  -sin u (du/dx)

dy/dx  =  -sin (a³ + x³) (3x²)

=  - 3x²sin (a³ + x³)

Question 5 :

Differentiate y  =  e^-mx

Solution :

y  =  e^-mx

Let u = -mx

Differentiate the function "u" with respect to x, we get

du/dx  =  -m (1)

=  -m

y  =  e^u

Differentiate the function "y" with respect to x, we get

dy/dx  =  e^u (du/dx)

dy/dx  =   e^-mx (-m)

dy/dx  =  -me^-mx

Question 6 :

Differentiate y  =  4 sec 5x

Solution :

y  =  4 sec 5x

Let u = 5x

Differentiate the function "u" with respect to x, we get

du/dx  =  5(1)

=  5

y  =  4 sec u

Differentiate the function "y" with respect to x, we get

dy/dx  =  4 sec u tan u (du/dx)

dy/dx  =  4 sec 5x tan 5x (5)

dy/dx  =  20 sec 5x tan 5x

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