If f(x) =xe^(5x+4)  and g(x) = cos2x , what is f'(g(x)) ?

Sep 16, 2016

$= {e}^{5 \cos 2 x + 4} \left(1 + 5 \cos 2 x\right)$

Explanation:

whilst the intention of this question may have been to encourage the use of the chain rule on both $f \left(x\right)$ and $g \left(x\right)$ - hence, why this is filed under Chain Rule - that is not what the notation asks for.

to make the point we look at the definition

$f ' \left(u\right) = \frac{f \left(u + h\right) - f \left(u\right)}{h}$

or

$f ' \left(u \left(x\right)\right) = \frac{f \left(u \left(x\right) + h\right) - f \left(u \left(x\right)\right)}{h}$

the prime means differentiate wrt to whatever is in the brackets

here that means, in Liebnitz notation: (d(f(x)))/(d(g(x))

contrast with this the full chain rule description:

$\left(f \setminus \circ g\right) ' \left(x\right) = f ' \left(g \left(x\right)\right) \setminus \cdot g ' \left(x\right)$

So, in this case, $u = u \left(x\right) = \cos 2 x$ and so the notation requires simply the derivative of $f \left(u\right)$ wrt to $u$, and then with $x \to \cos 2 x$, ie $\cos 2 x$ inserted as x in the resultant derivative

So here
 f'(cos 2x) qquad["let " u = cos 2x]

$= f ' \left(u\right)$

by the product rule
$= \left(u\right) ' {e}^{5 u + 4} + u \left({e}^{5 u + 4}\right) '$

$= {e}^{5 u + 4} + u \cdot 5 {e}^{5 u + 4}$

$= {e}^{5 u + 4} \left(1 + 5 u\right)$

So
$f ' \left(g \left(x\right)\right) =$f'(cos 2x)

$= {e}^{5 \cos 2 x + 4} \left(1 + 5 \cos 2 x\right)$

in short

$f ' \left(g \left(x\right)\right) \ne \left(f \setminus \circ g\right) ' \left(x\right)$

Sep 16, 2016

$f ' \left(g \left(x\right)\right) = {e}^{5 \cos \left(2 x\right) + 4} \left(1 + 5 \cos 2 x\right)$

Explanation:

$f \left(x\right) = x {e}^{5 x + 4}$
To find $f ' \left(g \left(x\right)\right)$, first we have to find $f ' \left(x\right)$ then we have to substitute $x$ by $g \left(x\right)$

$f ' \left(x\right) = {e}^{5 x + 4} + 5 x {e}^{5 x + 4}$
$f ' \left(x\right) = {e}^{5 x + 4} \left(1 + 5 x\right)$
Let us substitute $x$ by $f \left(x\right)$
$f ' \left(g \left(x\right)\right) = {e}^{5 \cos \left(2 x\right) + 4} \left(1 + 5 \cos 2 x\right)$