Question

We need to use the chain rule within the chain rule! For example, to evaluate\\ $\frac{d}{dx}\sin\left(e^{(5x^2+5x+4)}\right)$.\\ Let $f(x) = \sin(x)$ and $g(x) = e^{(5x^2+5x+4)}$. Then a first application of the chain rule gives\\ $\frac{d}{dx}\sin\left(e^{(5x^2+5x+4)}\right) = f'(g(x))g'(x)$.\\ It is possible at this stage to find\\ $f'(g(x)) = \cos\left(e^{(5x^2+5x+4)}\right)$\\ However to find $g'(x)$ requires a further use of the chain rule\\ $g'(x) = (10x+5)e^{(5x^2+5x+4)}$\\ Hence\\ $\frac{d}{dx}\sin\left(e^{(5x^2+5x+4)}\right) = (10x e^{(5x^2+5x+4)}\cos(e^{(5x^2+5x+4)})+5e^{(5x^2+5x+4)}\cos(e^{(5x^2+5x+4)}) $

          We need to use the chain rule within the chain rule! For example, to evaluate\\
$\frac{d}{dx}\sin\left(e^{(5x^2+5x+4)}\right)$.\\
Let $f(x) = \sin(x)$ and $g(x) = e^{(5x^2+5x+4)}$. Then a first application of the chain rule gives\\
$\frac{d}{dx}\sin\left(e^{(5x^2+5x+4)}\right) = f'(g(x))g'(x)$.\\
It is possible at this stage to find\\
$f'(g(x)) = \cos\left(e^{(5x^2+5x+4)}\right)$\\
However to find $g'(x)$ requires a further use of the chain rule\\
$g'(x) = (10x+5)e^{(5x^2+5x+4)}$\\
Hence\\
$\frac{d}{dx}\sin\left(e^{(5x^2+5x+4)}\right) = (10x e^{(5x^2+5x+4)}\cos(e^{(5x^2+5x+4)})+5e^{(5x^2+5x+4)}\cos(e^{(5x^2+5x+4)}) $

We need to use the chain rule within the chain rule! For example, to evaluate

(d)/(dx)sin(e^(5x^2+5x+4)).

Let f(x) = sin(x) and g(x) = e^(5x^2+5x+4). Then a first application of the chain rule gives

(d)/(dx)sin(e^(5x^2+5x+4)) = f'(g(x))g'(x).

It is possible at this stage to find

f'(g(x)) = cos(e^(5x^2+5x+4))

However to find g'(x) requires a further use of the chain rule

g'(x) = (10x+5)e^(5x^2+5x+4)

Hence

(d)/(dx)sin(e^(5x^2+5x+4)) = (10x e^(5x^2+5x+4)cos(e^(5x^2+5x+4))+5e^(5x^2+5x+4)cos(e^(5x^2+5x+4))

Added by Abigail L.

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