use chain rule to find the derivative of h(x) =(x+1) (x+2)(x+3).
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Answer :
dh/dx = 3x² + 12x + 11
Note :
If y = u•v , then
dy/dx = v•(du/dx) + u•(dv/dx)
Solution :
- Given : h(x) = (x + 1)(x + 2)(x + 3)
- To find : dh/dx = ?
We have ,
h(x) = (x + 1)(x + 2)(x + 3)
Now ,
Differentiating both the sides with respect to x , we have ;
=> dh/dx = d[(x + 1)(x + 2)(x + 3)]/dx
=> dh/dx = d[(x + 1)•(x + 2)(x + 3)]/dx
=> dh/dx = (x + 2)(x + 3)•d(x + 1)/dx
+ (x+1)•d[(x + 2)(x + 3)/dx
=> dh/dx = (x + 2)(x + 3)•1
+ (x+1)•d[(x + 2)•(x + 3)/dx
=> dh/dx = (x + 2)(x + 3) + (x+1)•[(x + 3)•d(x + 2)/dx + (x + 2)•d(x + 3)/dx]
=> dh/dx = (x + 2)(x + 3)
+ (x+1)•[(x + 3)•1 + (x + 2)•1]
=> dh/dx = (x + 2)(x + 3)
+ (x+1)•(x + 3 + x + 2)
=> dh/dx = (x + 2)(x + 3) + (x+1)•(2x + 5)
=> dh/dx = x²+3x+2x+6 + 2x²+ 5x + 2x + 5
=> dh/dx = 3x² + 12x + 11
Hence ,
dh/dx = 3x² + 12x + 11
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