Using the quotient rule, find the derivation of the following function:
a) (ax^2+b)/(cx+d)
Here is my work but i need help because am lit bit confuse
solution
a) (ax^2+b)/(cx+d)
df/dx = (2ax)(cx+d)-(ax^2+b) (c)/(cx+d)^2
df/dx = 2acx^2+2adx-acx^2+bc/(cx+d)^2
df/dx =acx^2+2adx+bc/(cx+d)^2
df/dx = c(ax^2+b)+ 2adx/(cx+d)^2
df/dx = ax^2+b+2ax/cx^2+d
[ bottom d top/dx - top d bottom/dx]/bottom^2
[(cx+d)(2ax) - (ax^2+b)(c)] /(cx+d)^2 agree
= [2acx^2+2adx - acx^2-bc] /(cx+d)^2
disagree sign of bc
= [acx^2 + 2adx - bc ] /(cx+d)^2
To find the derivative of the function (ax^2 + b)/(cx + d) using the quotient rule, follow these steps:
1. Identify the numerator (top) and denominator (bottom) of the function.
- Numerator: ax^2 + b
- Denominator: cx + d
2. Apply the quotient rule formula. The formula states that the derivative of a function f(x)/g(x) is given by [(f'(x)g(x) - f(x)g'(x))/[g(x)]^2].
3. Determine the derivative of the numerator, which is ax^2 + b.
- The derivative of ax^2 is 2ax (using the power rule).
- The derivative of b (a constant) is 0.
4. Determine the derivative of the denominator, which is cx + d.
- The derivative of cx is c (a constant) since d/dx(x) = 1.
- The derivative of d (a constant) is 0.
5. Substitute the derivatives into the quotient rule formula.
- Numerator: (2ax)(cx + d) - (ax^2 + b)(c)
- Denominator: (cx + d)^2
6. Expand and simplify the numerator and denominator.
- Numerator: 2acx^2 + 2adx - acx^2 - bc
- Denominator: (cx + d)^2
7. Combine like terms in the numerator.
- Numerator: (acx^2 - acx^2) + 2adx - bc
8. Simplify further.
- Numerator: 2adx - bc
9. Rewrite the final derivative expression.
- df/dx = (2adx - bc)/[(cx + d)^2]
So, the derivative of (ax^2 + b)/(cx + d) using the quotient rule is (2adx - bc)/[(cx + d)^2].