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### Course: AP®︎/College Calculus AB>Unit 2

Lesson 9: The product rule

# Differentiating products

We explore the product rule by finding the derivative of eˣcos(x). We identify eˣ as the first function and cos(x) as the second, then apply the product rule to calculate the derivative, simplifying our result for a clearer understanding.

## Want to join the conversation?

• Could someone help explain to me (or point me towards a video) why d/dx e^x = e^x? I haven't heard of this before :) Thanks!
• what if you have the product of 3 functions? like sinx*cosx*lnx.. etc
• Just apply the product rule twice:
(𝑓 ∙ 𝑔 ∙ ℎ)' = 𝑓 ' ∙ (𝑔 ∙ ℎ) + 𝑓 ∙ (𝑔 ∙ ℎ)' =
= 𝑓 ' ∙ 𝑔 ∙ ℎ + 𝑓 ∙ (𝑔' ∙ ℎ + 𝑔 ∙ ℎ') =
= 𝑓 ' ∙ 𝑔 ∙ ℎ + 𝑓 ∙ 𝑔' ∙ ℎ + 𝑓 ∙ 𝑔 ∙ ℎ'
• Why is d/dx of u(x)v(x) equal to u'(x)v(x)+u(x)v'(x) and not u'(x)v'(x)?
• When using the product rule, do both functions need to be functions of x (when I'm taking the derivative with respect to x)?
• Consider two functions, f(x) & g(y) where f only depends on x, and g only depends on y. Then d(f(x)*(g(y))/dx = df/dx * g(y) + dg/dx * f(x). Since g only depends on y, it is a constant w.r.t. x so dg/dx = 0. The two are equivalent when the variables are independent of each other. However, if they are not independent - the product rule will certainly still hold - we just must modify our definition of derivative a little bit using vectors instead of just real numbers.
• How do we know that v(x) = Cos x is equal to v'(x) = -Sin x? As in, is there a rule of which that tells the derivative of Cos, Sin or Tan?
(1 vote)
• So....product rule=power rule?
I'm totally confused. whether which method i use i should get the same answer right? but using the power rule you get e^x*-Sin(x), using product rule you get thing at
confused because now i don't know which method is right to use now
• Using the power rule gets you nothing, because the power rule only applies to polynomials. The power rule says that d/dx(xⁿ)=n·xⁿ⁻¹.

There is no rule that gives e^x·(-sin(x)) as the derivative of e^x·cos(x).
• Why is the 1st derivative of ln(x) = 1/x?
• Let's say that y=ln(x). Instead of taking the derivative of both sides here, we can re-write and differentiate implicitly. y=ln(x)>>>>x=e^y.
Hence, x=e^y
d/dx[x]=d/dx[e^y]
1=e^y*dy/dx (Chain rule)
dy/dx=1/e^y
dy/dx=1/x (Because x=e^y)
I hope this helped.
(1 vote)
• Can someone show me how to differentiate
y= 3x^2(4x+1)^3
Thanks
(1 vote)
• how does the derivative of tan^2x be shown in a graph
(1 vote)
• Doi you mean something like this?
http://www.wolframalpha.com/input/?i=graph+of+derivative+of+tan%5E2(x)

Wolfram alpha is a good site for generating quick graphs like that.