How do you differentiate f(x)=e^(5x^2+x+3) f(x)=e5x2+x+3 using the chain rule?

2 Answers
Feb 25, 2016

f'(x) = (10x + 1 )e^(5x^2+x+3)f'(x)=(10x+1)e5x2+x+3

Explanation:

Using the color(blue)" chain rule " chain rule

d/dx[f(g(x)) ] = f'(g(x)).g'(x) ddx[f(g(x))]=f'(g(x)).g'(x)

and d/dx(e^x) = e^xddx(ex)=ex

f'(x) = e^(5x^2+x+3) d/dx(5x^2 + x + 3)=e5x2+x+3ddx(5x2+x+3)

= e^(5x^2+x+3)(10x + 1 )=e5x2+x+3(10x+1)

Feb 25, 2016

h'(x)=e^(5x^2+x+3)(10x+1)h'(x)=e5x2+x+3(10x+1)

Explanation:

The given equation is h(x)=e^(5x^2+x+3)h(x)=e5x2+x+3. But it seems as if it is a function inside a function. So, let's write g(x)=5x^2+x+3g(x)=5x2+x+3 and f(x)=e^xf(x)=ex.
So that means f(g(x))=e^(g(x))=e^(5x^2+x+3f(g(x))=eg(x)=e5x2+x+3

From chain rule, we have d/dx(f(g(x))=h'(x)=f'(g(x))* g'(x)ddx(f(g(x))=h'(x)=f'(g(x))g'(x).
So, taking for f(g(x))f(g(x)) and g(x)g(x), we get
h'(x)=e^(g(x))*d/dx(g(x))==e^g(x)(10x+1)h'(x)=eg(x)ddx(g(x))==eg(x)(10x+1)

You can substitute it all to get back the proper answer.