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Differentiate the functions with respect to x: $ \; sin ( x^2 + 5 ) $

$\begin{array}{1 1} 2cos(x^2+5)\\ 2x.cos(x^2+5) \\ 2x.cos(x^2-5) \\ 2.cos(x^2+5)\end{array} $

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Toolbox:
  • According to the Chain Rule for differentiation, given two functions $f(x)$ and $g(x)$, and $y=f(g(x)) \rightarrow y' = f'(g(x)).g'(x)$.
  • $\; \large \frac{d(sinx)}{dx} $$= cosx$
Given $y = sin(x^2+5)$:
According to the Chain Rule for differentiation, given two functions $f(x)$ and $g(x)$, and $y=f(g(x)) \rightarrow y' = f'(g(x)).g'(x)$
$\Rightarrow$ Given $g(x) = x^2 + 5 \rightarrow g'(x) = 2x^{2-1} = 2x $
$\; \large \frac{d(sinx)}{dx} $$= cosx$
$\Rightarrow f'(g(x)) = f'(cos (x^2+5))$
Therefore $y' = f'(g(x)).g'(x) = cos(x^2+5). 2x = 2x.cos(x^2+5)$

 

answered Apr 4, 2013 by balaji.thirumalai
 
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