How do you find the derivative of #e^x * e^(2x) * e^(3x) * e^(4x)#?

Question

Paisley Johnson · Accepted Answer

#d/dx e^x*e^(2x)*e^(3x)*e^(4x) =10e^(10x)# Using the property that #a^x*a^y = a^(x+y)# along with that #d/dx e^x=e^x# and the chain rule, we have #d/dx e^x*e^(2x)*e^(3x)*e^(4x) = d/dx e^(x+2x+3x+4x)# #=d/dxe^(10x)# #=e^(10x)(d/dx10x)# (Using the chain rule with the functions #e^x# and #10x#) #=10e^(10x)#

Axel Alvarez · Answer

undefined To find the derivative of $ e^x \cdot e^{2x} \cdot e^{3x} \cdot e^{4x} $, you can use the product rule of differentiation. The product rule states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Applying this rule to the given expression, we get:

$$ \frac{d}{dx} (e^x \cdot e^{2x} \cdot e^{3x} \cdot e^{4x}) = e^x \cdot e^{2x} \cdot e^{3x} \cdot \frac{d}{dx}(e^{4x}) + e^x \cdot e^{2x} \cdot \frac{d}{dx}(e^{3x}) \cdot e^{4x} + e^x \cdot \frac{d}{dx}(e^{2x}) \cdot e^{3x} \cdot e^{4x} + \frac{d}{dx}(e^x) \cdot e^{2x} \cdot e^{3x} \cdot e^{4x} $$

$$ = e^{10x} + 3e^{9x} + 2e^{8x} + e^{7x} $$