AP Calculus BC Practice Quiz: Applying the Power Rule

Correct Answer: A

The power rule for derivatives states that if $f(x) = x^r$, then $f'(x) = rx^{r-1}$. For $f(x) = x^7$, we have $r=7$. Applying the rule, $f'(x) = 7x^{7-1} = 7x^6$.

Correct Answer: B

Using the power rule for a function of the form $f(x) = ax^r$, the derivative is $f'(x) = a \cdot rx^{r-1}$. For $g(x) = 5x^3$, we have $a=5$ and $r=3$. Therefore, $g'(x) = 5 \cdot 3x^{3-1} = 15x^2$.

Correct Answer: C

First, rewrite the function using a negative exponent: $h(x) = x^{-4}$. Now, apply the power rule where $r=-4$. The derivative is $h'(x) = -4x^{-4-1} = -4x^{-5}$. Rewriting this with a positive exponent gives $h'(x) = -\frac{4}{x^5}$.

Correct Answer: A

First, express the radical function using a fractional exponent: $y = x^{1/5}$. Now, apply the power rule with $r = 1/5$. The derivative is $\frac{dy}{dx} = \frac{1}{5}x^{(1/5) - 1} = \frac{1}{5}x^{(1/5) - (5/5)} = \frac{1}{5}x^{-4/5}$.

Correct Answer: A

The derivative of a sum is the sum of the derivatives. We apply the power rule to each term. The derivative of $4x^3$ is $4 \cdot 3x^{3-1} = 12x^2$. The derivative of $-7x$ (which is $-7x^1$) is $-7 \cdot 1x^{1-1} = -7x^0 = -7$. The derivative of a constant (2) is 0. Combining these, $f'(x) = 12x^2 - 7 + 0 = 12x^2 - 7$.

Correct Answer: A

Apply the power rule for $g(t) = at^r$, where $a=8$ and $r=3/2$. The derivative is $g'(t) = a \cdot rt^{r-1} = 8 \cdot \frac{3}{2}t^{(3/2)-1}$. Simplifying, $g'(t) = 12t^{(3/2)-(2/2)} = 12t^{1/2}$.

Correct Answer: B

First, rewrite the function in the form $ax^r$. The term $\sqrt{x}$ is $x^{1/2}$. Since it's in the denominator, the function is $f(x) = \frac{3}{2}x^{-1/2}$. Now, apply the power rule with $a = 3/2$ and $r = -1/2$. The derivative is $f'(x) = \frac{3}{2} \cdot (-\frac{1}{2})x^{(-1/2)-1} = -\frac{3}{4}x^{(-1/2)-(2/2)} = -\frac{3}{4}x^{-3/2}$.