Questions: Find the simplified difference quotient for the given function. f(x)=d x^2+ax+k The simplified difference quotient is

Solution Steps

Given the function: \[ f(x) = d x^2 + a x + k \]

Substitute \( x \) with \( x + h \) in the function: \[ f(x + h) = d (x + h)^2 + a (x + h) + k \] Expanding this, we get: \[ f(x + h) = d (x^2 + 2xh + h^2) + a x + a h + k \] \[ f(x + h) = d x^2 + 2 d x h + d h^2 + a x + a h + k \]

The difference quotient is given by: \[ \frac{f(x + h) - f(x)}{h} \] Substitute \( f(x + h) \) and \( f(x) \): \[ \frac{(d x^2 + 2 d x h + d h^2 + a x + a h + k) - (d x^2 + a x + k)}{h} \] Simplify the numerator: \[ \frac{d x^2 + 2 d x h + d h^2 + a x + a h + k - d x^2 - a x - k}{h} \] \[ \frac{2 d x h + d h^2 + a h}{h} \]

Factor out \( h \) from the numerator: \[ \frac{h (2 d x + d h + a)}{h} \] Cancel out \( h \): \[ 2 d x + d h + a \]

Final Answer