Questions: Attempt 1: 3 attempts remaining. Let f(x)=-7 x^3. Use the definition of derivative to find f'(x). f'(x)= Submit answer Next item

Solution Steps

To find the derivative of the function \( f(x) = -7x^3 \) using the definition of the derivative, we apply the limit definition:

\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

Substitute \( f(x) = -7x^3 \) into the definition and simplify the expression to find the derivative.

We start with the function \( f(x) = -7x^3 \). To find the derivative \( f'(x) \), we use the limit definition of the derivative:

\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

Substitute \( f(x) = -7x^3 \) and \( f(x+h) = -7(x+h)^3 \) into the limit definition:

\[ f'(x) = \lim_{h \to 0} \frac{-7(x+h)^3 + 7x^3}{h} \]

Expand \( (x+h)^3 \) and simplify the expression:

\[ (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \]

Substitute back:

\[ f'(x) = \lim_{h \to 0} \frac{-7(x^3 + 3x^2h + 3xh^2 + h^3) + 7x^3}{h} \]

Simplify:

\[ f'(x) = \lim_{h \to 0} \frac{-7x^3 - 21x^2h - 21xh^2 - 7h^3 + 7x^3}{h} \]

\[ f'(x) = \lim_{h \to 0} \frac{-21x^2h - 21xh^2 - 7h^3}{h} \]

Factor out \( h \) from the numerator:

\[ f'(x) = \lim_{h \to 0} \frac{h(-21x^2 - 21xh - 7h^2)}{h} \]

Cancel \( h \):

\[ f'(x) = \lim_{h \to 0} (-21x^2 - 21xh - 7h^2) \]

Evaluate the limit as \( h \to 0 \):

\[ f'(x) = -21x^2 \]

Final Answer