Derivative Rules Practice

These questions follow Derivative Rules — the power rule and its reasonable companions, the product rule, and slopes on demand. Fluency is the goal; speed will arrive on its own, uninvited.

Power, sum, and constant multiple

1. Verify the power rule for $f(x)=x^4$ using the definition of the derivative.
2. Differentiate $f(x)=5x^3-4x+7$, and determine the instantaneous rate of change at $x=-1$.
3. Differentiate $f(x)=\sqrt{x}$, and determine the slope of the tangent at $x=9$.

Answer 1

Expand $(x+h{)}^4=x^4+4x^{3}h+6x^2{h}^2+4x{h}^3+h^4$, so $\frac{(x+h{)}^4-x^4}{h}=4x^3+6x^{2}h+4x{h}^2+h^3$ As $h\to 0$, everything carrying an $h$ vanishes: $f^{\prime }(x)=4x^3$. The exponent hopped down front, the power dropped by one — the conjectured pattern, certified.

Answer 2

Term by term: $f^{\prime }(x)=15x^2-4+0$. The constant 7 contributes nothing — flat pieces have no rate. At $x=-1$: $f^{\prime }(-1)=15(1)-4=11$.

Answer 3

Rewrite as a power: $f(x)=x^{1/2}$, so $f^{\prime }(x)=\frac{1}{2}{x}^{-1/2}=\frac{1}{2\sqrt{x}}$. At $x=9$: $f^{\prime }(9)=\frac{1}{2\times 3}=\frac{1}{6}$. Reasonable? The square root curve is climbing but flattening at $x=9$ — a small positive slope is exactly right.

The product rule

4. Differentiate $f(x)=(3x+2)(2x^2-1)$ twice: once with the product rule, once by expanding first. Confirm the answers agree.
5. Differentiate $f(x)=(x^2+1)(x^3-2x)$ using the product rule.

Answer 4

Product rule: $f^{\prime }(x)=3(2x^2-1)+(3x+2)(4x)=6x^2-3+12x^2+8x$, which is $18x^2+8x-3$. Expanding first: $f(x)=6x^3+4x^2-3x-2$, so $f^{\prime }(x)=18x^2+8x-3$. The two roads agree — and that agreement is the reason to trust the product rule on functions you cannot expand.

Answer 5

$f^{\prime }(x)=2x(x^3-2x)+(x^2+1)(3x^2-2)$ $=2x^4-4x^2+3x^4-2x^2+3x^2-2=5x^4-3x^2-2$ Audit by expanding: $f(x)=x^5-x^3-2x$ gives $f^{\prime }(x)=5x^4-3x^2-2$. ✓

Slopes on demand

6. Determine the derivative of $f(x)=2x^3+3x^2$, and the point(s) on the graph where the slope of the tangent is 36.
7. Determine the equation of the tangent line to $f(x)=x^3-3x^2+2$ at $x=1$.
8. Water flows into two barrels. The volumes in litres after $t$ minutes are $f(t)$ and $g(t)$. Explain what $f^{\prime }(t)$, $g^{\prime }(t)$, $f^{\prime }(t)+g^{\prime }(t)$, and $(f+g{)}^{\prime }(t)$ each represent, and how this story verifies the sum rule.

Answer 6

$f^{\prime }(x)=6x^2+6x$. Set it to 36: $6x^2+6x-36=0$, so $x^2+x-6=(x+3)(x-2)=0$, giving $x=-3$ and $x=2$. The points: $f(-3)=-54+27=-27$ and $f(2)=16+12=28$, so $(-3,-27)$ and $(2,28)$. Two points — a cubic’s slopes repeat, once on each arm.

Answer 7

Point: $f(1)=1-3+2=0$, so $(1,0)$. Slope: $f^{\prime }(x)=3x^2-6x$, so $f^{\prime }(1)=3-6=-3$. Line through $(1,0)$ with slope $-3$: $y=-3(x-1)$, or $y=-3x+3$.

Answer 8

$f^{\prime }(t)$ and $g^{\prime }(t)$ are the flow rates into each barrel, in litres per minute. Their sum $f^{\prime }(t)+g^{\prime }(t)$ is the combined inflow measured barrel by barrel; $(f+g{)}^{\prime }(t)$ is the rate the total volume grows, measured on the total. Water does not care how you account for it — both describe the same litres arriving per minute, so they must be equal. That physical certainty is the sum rule: $(f+g{)}^{\prime }=f^{\prime }+g^{\prime }$.