How do you differentiate #f(x)=4x*(x-4)*tanx# using the product rule?

Question

Ezra Adams · Accepted Answer

#f'(x) = 8(x-2)(tanx) + 4x(x-4)(sec^2x)#

#f(x) = 4(x)(x-4)(tanx)#

The product rule states that #color(blue)(d/dx (u*v*w) = (u)'vw + u(v)'w + uv(w)')#

I'm going to distribute the polynomial terms to take a shortcut: #f(x) = 4(x^2-4x)(tanx)#

#f'(x) = 4(x^2-4x)'(tanx) + 4(x^2-4x)(tanx)'#

#f'(x) = 4(2x-4)(tanx) + 4(x^2-4x)(sec^2 x)#

Simplifying by factoring:

(Note: don't try substituting the trig identity #sec^2x = tan^2 x + 1#. It just complicates the terms.)

#f'(x) = 8(x-2)(tanx) + 4x(x-4)(sec^2x)#

Paisley Johnson · Answer

# f'(x) = 8(x-2)tanx + 4x(x-4)sec^2x #

We have: # f(x) = 4x(x-4)tanx # We will apply the Product Rule for Differentiation: # d/dx(uv)=u(dv)/dx+(du)/dxv #, or, # (uv)' = (du)v + u(dv) # I was taught to remember the rule in words; "The first times the derivative of the second plus the derivative of the first times the second ". This can be extended to three products: # d/dx(uvw)=uv(dw)/dx+u(dv)/dxw + (du)/dxvw# # { ("Let", u = 4x, => , (du)/dx = 4), ("And" ,v =x-4, =>, (dv)/dx = 1 ), ("And", w = tanx, =>, (dw)/dx = sec^2x ) :}# Then: # d/dx(uvw) = (du)/dxvw + u(dv)/dxw+ + uv(dw)/dx# Gives us: # f'(x) = (4)(x-4)tanx + (4x)(1)tanx+ + 4x(x-4)sec^2x# # " " = 4{(x-4)tanx + xtanx + x(x-4)sec^2x} # # " " = 4{2xtanx-4tanx + x(x-4)sec^2x} # # " " = 4{(2x-4)tanx + x(x-4)sec^2x} # # " " = 8(x-2)tanx + 4x(x-4)sec^2x #

Adam Adkins · Answer

undefined To differentiate $ f(x) = 4x(x - 4)\tan(x) $ using the product rule:

1. Identify the two functions being multiplied: $ u(x) = 4x(x - 4) $ and $ v(x) = \tan(x) $.
2. Apply the product rule formula: $ (uv)' = u'v + uv' $.
3. Differentiate each function:
   $ u'(x) $ (derivative of $ u(x) $) = $ (4x)'(x - 4) + 4x(x - 4)' $
   $ v'(x) $ (derivative of $ v(x) $) = $ \sec^2(x) $ (derivative of $ \tan(x) $).
4. Substitute the derivatives and the original functions into the product rule formula.
   $ f'(x) = (4(x - 4) + 4x) \tan(x) + 4x(x - 4) \sec^2(x) $.
5. Simplify the expression, if necessary, to get the final result.