To find the expression \(\frac{f(x + h) - f(x)}{h}\) for the function \(f(x) = 4x^2 + 5\), we will first calculate \(f(x + h)\) and then substitute back into the expression.
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Calculate \(f(x + h)\): \[ f(x + h) = 4(x + h)^2 + 5 \]
Now, expand \((x + h)^2\): \[ (x + h)^2 = x^2 + 2xh + h^2 \]
So, substituting this back into \(f(x + h)\): \[ f(x + h) = 4(x^2 + 2xh + h^2) + 5 = 4x^2 + 8xh + 4h^2 + 5 \]
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Now, calculate \(f(x + h) - f(x)\): \[ f(x + h) - f(x) = (4x^2 + 8xh + 4h^2 + 5) - (4x^2 + 5) \]
The \(4x^2 + 5\) terms cancel out: \[ f(x + h) - f(x) = 8xh + 4h^2 \]
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Now put it into the expression \(\frac{f(x + h) - f(x)}{h}\): \[ \frac{f(x + h) - f(x)}{h} = \frac{8xh + 4h^2}{h} \]
Now simplify by dividing each term by \(h\) (assuming \(h \neq 0\)): \[ = 8x + 4h \]
Thus, the resulting expression is: \[ \frac{f(x + h) - f(x)}{h} = 8x + 4h \]