document.write( "Question 274722: Hi, pls help me find derivative of following:\r
\n" ); document.write( "\n" ); document.write( "1. f(x) =16x+19, find f'(8)=?
\n" ); document.write( "2. if f(x)= 2e^(x+1) + e^1, then f'(0) = ?
\n" ); document.write( "2. g(x)=(e^x)/(1+3x), find g'(x)=?
\n" ); document.write( "4. if f(t)= sq.root of 5/t^7, find f'(t)=?\r
\n" ); document.write( "\n" ); document.write( "Thank you, \n" ); document.write( "
Algebra.Com's Answer #200652 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
We'll need the following to solve your problems:
    In each of the these, c and n are constants and f, g, u and v are functions
  1. if f(x) = c then f'(x) = 0
  2. if f = x then f' = 1
  3. if f = c*g then f' = c*g'
  4. if f = u + v then f' = u' + v'
  5. if \"f+=+u%5En\" then f' = \"n%2Au%5E%28n-1%29\"*u'. Rule 5a: if \"f+=+x%5En\", then f' = \"n%2Ax%5E%28n-1%29\"
  6. if f = u/v then f' = (v*u' - u*v')/v^2
  7. if \"f+=+e%5Eu\" then f' = u'*\"e%5Eu\"

\n" ); document.write( "To find derivatives you need to learn how to take a function and figure out how to express it in the form of one or more of the rules for derivatives. (Note: I've only listed the ones we need for these problems. There are more.)
\n" ); document.write( "1. f(x) =16x+19, find f'(8)=?
\n" ); document.write( "Let u(x) = 16x and v(x) = 19, which makes f = u + v. Then by rule #4:
\n" ); document.write( "f' = u' + v'
\n" ); document.write( "By rule #1 v' = 0 so now
\n" ); document.write( "f' = u'
\n" ); document.write( "For u', let c = 16 and g(x) = x. By rule #3 u' = c*g' so now
\n" ); document.write( "f' = 16*g'
\n" ); document.write( "From rule #2, g' = 1 so now
\n" ); document.write( "f' = 16*1 = 16
\n" ); document.write( "Since f' is a constant, f'(anything, including 8) = 16
\n" ); document.write( "I hope this makes sense. After all, f(x) is a line with a (constant) slope of 16 and first derivatives are formulas for slope. We will used this in the problems that follow. For any function that is a line we will just use the coefficient of x (the slope) for the derivative instead of working through all these rules for derivatives.

\n" ); document.write( "2. if \"f%28x%29=+2e%5E%28x%2B1%29+%2B+e%5E1\", then f'(0) = ?
\n" ); document.write( "Let \"u%28x%29+=+2e%5E%28x%2B1%29\" and \"v%28x%29+=+e%5E1\" which makes f = u + v. By rule #4:
\n" ); document.write( "f' = u' + v'
\n" ); document.write( "Since v(x) is a constant (\"e%5E1\" is simply a number, after all.) then v' = 0 (rule #1) so now
\n" ); document.write( "f' = u'
\n" ); document.write( "Let c = 2 and \"g%28x%29+=+e%5E%28x%2B1%29\" which makes u = c*g. By rule #3, u' = c*g' so now
\n" ); document.write( "f' = 2*g'
\n" ); document.write( "Let h(x) = x+1 which makes \"g%28x%29+=+e%5Eh\". By rule #7 g' = h'*\"e%5Eh\". Since h is a line, h' is the slope of that line: 1 which means g' = \"e%5Eh\". So now
\n" ); document.write( "f' = \"2%2Ae%5Eh+=+2%2Ae%5E%28x%2B1%29\"
\n" ); document.write( "f'(0) = \"2e%5E%280%2B1%29+=+2e%5E1+=+2e\"

\n" ); document.write( "3. \"g%28x%29=%28e%5Ex%29%2F%281%2B3x%29\", find g'(x)=?
\n" ); document.write( "Let u(x) = \"e%5Ex\" and v(x) = 1+3x which makes g = u/v. By rule #6
\n" ); document.write( "g' = (v*u' - u*v')/v^2
\n" ); document.write( "By rules 7 and 2, u' = \"e%5Ex\". And since v is a line, v' is the slope of that line: 3. So now
\n" ); document.write( "g'(x) =

\n" ); document.write( "4. if \"f%28t%29=+sqrt%285%2Ft%5E7%29\", find f'(t)=?
\n" ); document.write( "To make this one a little easier I'm going to rewrite f(t) using exponents:
\n" ); document.write( "
\n" ); document.write( "(This was not necessary but it does make the derivative much simpler.)
\n" ); document.write( "Let c = \"sqrt%285%29\" and \"g%28t%29+=+t%5E%28-7%2F2%29\" then by rule #3
\n" ); document.write( "f' = c*g'
\n" ); document.write( "By rule #5a, g' = \"%28-7%2F2%29t%5E%28-7%2F2+-+1%29+=+%28-7%2F2%29t%5E%28-9%2F2%29\". This makes
\n" ); document.write( "f'(t) = \"sqrt%285%29%2A%28-7%2F2%29t%5E%28-9%2F2%29+=+%28%28-7sqrt%285%29%29%2F2%29t%5E%28-9%2F2%29\"
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