If #f(x)= - e^(5x # and #g(x) = 2x^3 #, how do you differentiate #f(g(x)) # using the chain rule?

Question

If #f(x)=  -  e^(5x # and #g(x)  =  2x^3     #, how do you differentiate #f(g(x))  # using the chain rule?

William Abdalla · Accepted Answer

#(f(g(x)))'= -30x^2e^(10x^3)# #f(g(x))# is a composite function, so differentiating it is #" "# determined by applying chain rule. #" "# Chain rule : #" "# #(f(g(x)))'= f'(g(x))xxg'(x)# #" "# #" "# Let us compute #" "color(blue)(f'(g((x)))# #" "# Knowing the differentiation of #e^u#: #" "# #(e^u)' = u'e^u# #" "# #f'(x)=-5e^(5x)" " # #" "# then #" "# #f'(g(x)))=-5e^(5(g(x)))# #" "# #color(blue)(f'(g(x)))=-5e^(5(2x^3))# #" "# #color(blue)(f'(g(x))=-5e^(10x^3))# #" "# #" "# Differentiation of #g(x)# #" "# #g'(x)# is determined by applying the power rule. #" "# Power Rule: #" "# #color(red)((x^n)' = nx^(n-1))# #" "# #g'(x)=2(x^3)'=2(color(red)(3x^2))=6x^2# #" "# Therefore , #" "# #(f(g(x)))'= f'(g(x))xxg'(x)# #" "# #(f(g(x)))'= -5e^(10x^3)xx6x^2# #" "# #(f(g(x)))'= -30x^2e^(10x^3)#

Adam Adkins · Answer

undefined To differentiate $ f(g(x)) $ using the chain rule, follow these steps:

1. Substitute $ g(x) $ into $ f(x) $, so you get $ f(g(x)) = -e^{5g(x)} $.
2. Differentiate $ f(g(x)) $ with respect to $ x $ using the chain rule:
$$ \frac{d}{dx}[f(g(x))] = \frac{d}{dx}[-e^{5g(x)}] $$
$$ = -5e^{5g(x)} \cdot \frac{d}{dx}[g(x)] $$
3. Find the derivative of $ g(x) $ with respect to $ x $.
4. Differentiate $ g(x) $ with respect to $ x $:
$$ \frac{d}{dx}[g(x)] = \frac{d}{dx}[2x^3] $$
$$ = 6x^2 $$
5. Substitute the derivative of $ g(x) $ back into the expression:
$$ \frac{d}{dx}[f(g(x))] = -5e^{5g(x)} \cdot 6x^2 $$
6. Simplify the expression if necessary.

So, the derivative of $ f(g(x)) $ with respect to $ x $ using the chain rule is:
$$ \frac{d}{dx}[f(g(x))] = -30x^2e^{5g(x)} $$