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I am confused on finding the derivative when constants are involved: with $3\ln(x^4 + \sec x)$ the derivative is $(3(4x^3 + \sec x \tan x))/(x^4+\sec x)$ notice the three stayed, but with

$x^2+2x+7$ the derivative of seven turns to zero in the answer $2(x+1)$

my question is what is the difference.

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4 Answers 4

up vote 1 down vote accepted

We have that, for any differentiable $f$

$$(f+c)'=f'+0=f'$$

however, $$(c\cdot f)'=c\cdot f'$$

for any constant $c$. That should probably be in your notes.

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Thanks Peter!!! I should have caught that!!! –  codenamejupiterx Mar 13 '13 at 2:14
    
What is you have d/dx of (1-cx)^2 –  Conrad C Oct 21 at 18:16

The difference is that there is no constant term in the first expression. $3$ there is coefficient, not a constant term!

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OK! Thanks Shu Xiao Li! –  codenamejupiterx Mar 13 '13 at 2:11

Remember that $$\frac d {dx} (f(x) +g(x)) = \frac d {dx} (f(x)) + \frac d {dx} (g(x))$$ but $$\frac d {dx} (cf(x)) = c \frac d {dx} (f(x)),$$

if $c$ is a constant.

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As 3 is constant with in multiplication with function therefore 3 can taken outside of differentiation by the rule d/dx(kf(x)) = kd/dx(f(x)) while in second example 7 is constant therefore its differentiation is zero due to rule d/dx(C) = 0 where c = constant

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