Answer to Question #203444 in Calculus for jay

Question

Given that 𝑓(𝑥) = 3𝑥 2 − 4𝑥 + 7, use the definition of the derivative to find 𝑓 ′ (𝑥)

Answer 1

Solution:

$𝑓(𝑥) = 3𝑥^2 − 4𝑥 + 7$

By definition of derivatives:

$\begin{aligned} f^{\prime}(x) &=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\ &=\lim _{h \rightarrow 0} \frac{\left(3(x+h)^{2}-4(x+h)+7\right)-\left(3 x^{2}-4 x-7\right)}{h} \\ &=\lim _{h \rightarrow 0} \frac{\left(3\left(x^{2}+2 x h+h^{2}\right)-4 x-4 h\right)-\left(3 x^{2}-4 x\right)}{h} \\ &=\lim _{h \rightarrow 0} \frac{3 x^{2}+6 x h+3 h^{2}-4 x-4 h-3 x^{2}+4 x}{h} \\ &=\lim _{h \rightarrow 0} \frac{6 x h+3 h^{2}-4 h}{h} \\ &=\lim _{h \rightarrow 0}(6 x+3 h-4) \\ &=6 x-4 \end{aligned}$