Derivatives Of Polynomials & Exponential

Question 1

Find the 1st Derivative of f(x)

Answer 1

-Multiply exponent by the coefficient of x
-Subtract 1 from the exponent
-Move denominators to the numerator and change the sign of the exponent to -
-then multiply that exponent by the coefficient of its term and -1 from the exponent
** the first derivative changes any square roots to fractions - ex. 32(sqrt(x)) becomes 16x^1/2

Question 2

The SECOND Derivative

Answer 2

-Repeat steps for first derivative and after you get your second round of derivative numbers by multiplying the exponents by the coefficients and subtracting 1 from the exponents again,
NOW put any negative exponents back into the denominator making the new negative exponent you calculated positive again by putting it back in the denominator.
(first derivative removes fractions, second derivative puts them back into place!)

Question 3

Lim[ C*f(x) ] =

The limit of a Constant * a Function= ?

Answer 3

Lim[ Cf(x) ] = Clim f(x)

= the Constant * the limit of the function

Question 4

The Derivative of a Function with Radicals

Answer 4

-to write a radical as a fraction, the exponent becomes the numerator and the index becomes the denominator of the power that base # is raised to.
- for the sqrt(x), the index is 2 and the exponent is 1–> x^1/2
Cube root, the index is 3, exponent is 1–> x^1/3
—When the bases are the same, you just add the exponents together, for example 2x^4x^1/2 =2x1x=2x & 4+1/2 =9/2 (4*2+1=9 over the 2) put it together we have 2x^9/2
ALWAYS REMOVE ALL FRACTIONS FIRST BUT MOVING DENOMINATORS UP WITH NEGATIVE EXPONENTS
—THEN multiply your exponents by your coefficients and subtract 1 from your exponents on all terms,
—LAST, Rewrite your terms removing the negative exponents by moving them back to the denominator, for example (28/3)x^-10/3 becomes 28/(3x^10/3)

Question 5

To put the Derivative of a function back in Radical Form

Ex. F’(x)=9x^7/2 + 28/(3x^10/3)

Becomes
f’(x)=9x^3sqrt(x)+ 28/(3x^3cuberoot(x))

Answer 5

F’(x)= 9sqrt(x^7) + 28/(3cuberoot(x^10)
And simply further
9x^3*sqrt(x)

Sqrt(x^7) splits to sqrt(x^6)sqrt(x) which becomes x^3sqrt(x)

28/3cuberoot(x^10) becomes 28/(3x^3the cube root of x)

Question 6

Lim [ f(x) + g(x) ] =

Answer 6

Lim [ f(x) + g(x) ] = lim f(x) + lim g(x)
x—>a x—>a

Question 7

Find the derivative of f(x)=4sin(x)

Answer 7

1. Find the derivative of f(x)=4sin(x)

Question 8

Lim [ f(x)*g(x) ]= ?
x—>a

Answer 8

Lim [ f(x)*g(x) ]= lim f(x) * lim g(x)
x—>a x—>a x—>a

Question 9

Lim [ f(x) ]^n =

Answer 9

Lim [ f(x) ]^n = n*lim f(x) provided the limit is defined