Differentiation Using Chain Rule Examples :
Here we are going to see how to find the derivatives using chain rule.
Question 1 :
Differentiate y = (x^{2} + 4x + 6)^{5}
Solution :
y = (x^{2} + 4x + 6)^{5}
Let u = x^{2} + 4x + 6
Differentiate the function "u" with respect to x, we get
du/dx = 2x + 4(1) + 0
= 2x + 4
y = u^{5}
Differentiate the function "y" with respect to x, we get
dy/dx = 5u^{4} (du/dx)
= 5(x^{2} + 4x + 6)^{4} (2x + 4)
Question 2 :
Differentiate y = tan 3x
Solution :
y = tan 3x
Let u = 3x
Differentiate the function "u" with respect to x, we get
du/dx = 3 (1)
= 3
y = tan u
Differentiate the function "y" with respect to x, we get
dy/dx = sec^{2}u (du/dx)
= sec^{2}3x (3)
= 3 sec^{2}3x
Question 3 :
Differentiate y = cos (tan x)
Solution :
y = cos (tan x)
Let u = tan x
Differentiate the function "u" with respect to x, we get
du/dx = sec^{2} x
y = cos u
Differentiate the function "y" with respect to x, we get
dy/dx = -sin u (du/dx)
= -sin (tan x) sec^{2}x
Question 4 :
Differentiate y = ∛(1 +x^{3})
Solution :
y = ∛(1 +x^{3})
Let u = 1 +x^{3}
Differentiate the function "u" with respect to x, we get
du/dx = 0 + 3x^{2}
y = u^{1/3}
Differentiate the function "y" with respect to x, we get
dy/dx = (1/3) u^{-2/3 }(du/dx)
= (1/3) (1 + x^{3})^{-2/3 }(3x^{2})
= x^{2}(1 + x^{3})^{-2/3}
Question 5 :
Differentiate y = e^{√x}
Solution :
y = e^{√x}
Let u = √x
Differentiate the function "u" with respect to x, we get
du/dx = 1/2√x
y = e^{u}
Differentiate the function "y" with respect to x, we get
dy/dx = e^{u }(du/dx)
= e^{√x}^{ }(1/2√x)
= e^{√x}/2√x
Question 6 :
Differentiate y = sin (e^{x})
Solution :
y = sin (e^{x})
Let u = e^{x}
Differentiate the function "u" with respect to x, we get
du/dx = e^{x}
y = sin u
Differentiate the function "y" with respect to x, we get
dy/dx = cos u (du/dx)
= cos (e^{x}) (e^{x})
= e^{x} cos (e^{x})
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