Use the definition of the derivative to find the derivative of the function. $f(x)=7 x^{4}-4 x+6$
A) $-4 x+6$
B) $-4 x^{3}+6$
C) $7 x^{3}-4$
D) $28 x^{3}-4$

Question

Quizplus · Accepted Answer

Using the definition of the derivative, we have

\begin{align*}
f'(x)&=\lim_{h 	o 0} \frac{f(x+h)-f(x)}{h}\
&=\lim_{h 	o 0} \frac{(7(x+h)^4-4(x+h)+6)-(7x^4-4x+6)}{h}\
&=\lim_{h 	o 0} \frac{7(x^4+4x^3h+6x^2h^2+4xh^3+h^4)-4x-4h+6-7x^4+4x-6}{h}\
&=\lim_{h 	o 0} \frac{28x^3h+42x^2h^2+16xh^3+7h^4}{h}\
&=\lim_{h 	o 0} 28x^3+42x^2h+16xh^2+7h^3\
&=28x^3-4 
\end{align*}

Therefore, the correct choice is D.

Use the Definition of the Derivative to Find the Derivative f(x)=7x4−4x+6f(x)=7 x^{4}-4 x+6f(x)=7x4−4x+6

Use the Definition of the Derivative to Find the Derivative $f(x)=7 x^{4}-4 x+6$