Question B.4: Use Equation B.28 to find the derivative of the following fu...
Use Equation B.28 to find the derivative of the following function: y(x) = ax³ + bx + c, where a, b, and c are constants.
\frac{d y}{d x}=\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}=\lim _{\Delta x \rightarrow 0} \frac{y(x+\Delta x)-y(x)}{\Delta x} (B.28)
Evaluate the function at x + Δx:
y(x + Δx) = a(x + Δx)³ + b(x + Δx) + c
= a(x³ + 3x² Δx + 3x Δx² + Δx³) + b(x + Δx) + c
Evaluate the numerator of Equation B.28:
Δy = y(x + Δx) – y(x) = a(3x² Δx + 3x Δx² + Δx³) + b Δx
Substitute into Equation B.28 and take the limit:
\begin{aligned}& \frac{d y}{d x}=\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}=\lim _{\Delta x \rightarrow 0}\left[a\left(3 x^2+3 x \Delta x+\Delta x^2\right)\right]+b \\& \frac{d y}{d x}=3 a x^2+b\end{aligned}Related Answered Questions
Question: B.10
Verified Answer:
Because the result is determined by raising a quan...
Question: B.9
Verified Answer:
Because the result is a subtraction, add the absol...
Question: B.8
Verified Answer:
Because the result is a multiplication, add the pe...
Question: B.7
Verified Answer:
Write the quotient as g h^{-1} and ...
Question: B.6
Verified Answer:
Rewrite the function as a product:
y(x)=x^3...
Question: B.5
Verified Answer:
Apply Equation B.29 to each term separately and re...
Question: B.3
Verified Answer:
(A) Use the Pythagorean theorem:
c² = a² + b² = 2....
Question: B.2
Verified Answer:
Solve Equation (2) for x:
(3) x = y + 2
Substitu...
Question: B.1
Verified Answer:
Use Equation B.8 to find the roots:
x=\frac...