Differentiation Rules: practice solutions, College – download the PDF

1 Product rule on polynomials ★★★

Take \( u=x^2+1 \), \( v=x^3-2x \); then \( u'=2x \) and \( v'=3x^2-2 \).

\( f'(x)=2x(x^3-2x)+(x^2+1)(3x^2-2)=2x^4-4x^2+3x^4-2x^2+3x^2-2=5x^4-3x^2-2. \)

Check: \( f(x)=x^5-2x^3+x^3-2x=x^5-x^3-2x \), whose derivative is \( 5x^4-3x^2-2 \). Both methods agree.

2 A first quotient ★★★

\( g'(x)=\dfrac{1\cdot(x+3)-x\cdot 1}{(x+3)^2}=\dfrac{3}{(x+3)^2}. \)

At \( x=1 \): \( g'(1)=\dfrac{3}{16} \).

3 Power of a linear function ★★★

Outer: \( u^5 \); inner: \( u=4x-7 \) with \( u'=4 \).

\( h'(x)=5(4x-7)^4\cdot 4=20(4x-7)^4. \) At \( x=2 \), \( 4x-7=1 \), so \( h'(2)=20 \).

4 Trigonometric sum ★★★

\( y'=3\cos x+2\sin x+\sec^2 x. \)

At \( x=0 \): \( 3(1)+2(0)+1=4 \).

5 Exponentials and logarithms ★★★

\( f'(x)=5e^x-\dfrac{4}{x}+2^x\ln 2. \)

\( f'(1)=5e-4+2\ln 2\approx 13.59-4+1.39=10.98. \)

6 Successive derivatives ★★★

\( f'(x)=4x^3-9x^2+2 \); \( f''(x)=12x^2-18x \); \( f^{(3)}(x)=24x-18 \); \( f^{(4)}(x)=24 \).

\( f''(2)=12(4)-18(2)=48-36=12. \)

7 Velocity and acceleration of a cart ★★★

Velocity: \( v(t)=s'(t)=3t^2-12t+9 \). Acceleration: \( a(t)=s''(t)=6t-12 \).

At \( t=4 \): \( v=48-48+9=9 \) ft/s and \( a=24-12=12 \) ft/s\( ^2 \).

At rest when \( 3(t^2-4t+3)=3(t-1)(t-3)=0 \), that is at \( t=1 \) s and \( t=3 \) s.

8 Product with a sine ★★★

\( f'(x)=2x\sin x+x^2\cos x. \)

At \( x=\pi \): \( 2\pi(0)+\pi^2(-1)=-\pi^2\approx -9.87. \)

9 Where is the tangent horizontal? ★★★

\( y'=\dfrac{e^x(x^2+1)-e^x(2x)}{(x^2+1)^2}=\dfrac{e^x(x^2-2x+1)}{(x^2+1)^2}=\dfrac{e^x(x-1)^2}{(x^2+1)^2}. \)

The denominator is never 0 and \( e^x\gt 0 \), so \( y'=0 \) only when \( x=1 \). The point is \( \left(1,\dfrac{e}{2}\right) \). Since \( y'\ge 0 \) everywhere, the function never decreases.

10 Cosine of a quadratic ★★★

Outer: \( \cos u \); inner: \( u=3x^2+1 \), \( u'=6x \). So \( y'=-\sin(3x^2+1)\cdot 6x=-6x\sin(3x^2+1) \).

At \( x=0 \): \( y'=0 \). At \( x=1 \): \( y'=-6\sin 4\approx -6(-0.7568)\approx 4.54. \)

11 Logarithm of a quadratic ★★★

\( x^2+4x+5=(x+2)^2+1\ge 1\gt 0 \), so the logarithm is defined for all real \( x \).

\( y'=\dfrac{2x+4}{x^2+4x+5} \). This is 0 only when \( 2x+4=0 \), that is \( x=-2 \).

12 Tangent line to an exponential product ★★★

\( f'(x)=e^{-2x}+x(-2)e^{-2x}=(1-2x)e^{-2x}. \)

(a) \( f(0)=0 \) and \( f'(0)=1 \), so the tangent is \( y=x \).

(b) \( f'(x)=0 \) when \( 1-2x=0 \), so \( x=\tfrac12 \). The point is \( \left(\tfrac12,\dfrac{1}{2e}\right)\approx(0.5,\,0.18) \).

13 Implicit slope of a tilted curve ★★★

Check: \( 4+2+1=7 \). Differentiate: \( 2x+y+x\,y'+2y\,y'=0 \), so \( y'(x+2y)=-(2x+y) \) and \( y'=-\dfrac{2x+y}{x+2y} \).

At \( (2,1) \): \( y'=-\dfrac{5}{4} \). Tangent: \( y-1=-\tfrac54(x-2) \), i.e. \( y=-\tfrac54x+\tfrac72 \).

14 Derivatives of x ln x ★★★

\( f'(x)=\ln x+x\cdot\dfrac1x=\ln x+1 \); \( f''(x)=\dfrac1x \); \( f^{(3)}(x)=-\dfrac{1}{x^2} \).

\( f''(4)=\dfrac14 \) and \( f^{(3)}(2)=-\dfrac14 \).

15 Find the mistakes ★★★

(i) The student differentiated only \( x^2 \) and ignored the product rule. Correct: \( 2x\,e^x+x^2e^x=x(x+2)e^x \).

(ii) The inner function \( 4x \) was ignored (chain rule). Correct: \( \cos(4x)\cdot 4=4\cos(4x) \).

16 A variable exponent ★★★

\( \ln y=\sin x\cdot\ln x \). Differentiating: \( \dfrac{y'}{y}=\cos x\ln x+\dfrac{\sin x}{x} \).

So \( y'=x^{\sin x}\left(\cos x\ln x+\dfrac{\sin x}{x}\right) \).

At \( x=\tfrac\pi2 \): \( y=\tfrac\pi2 \), \( \cos=0 \), \( \sin=1 \), so \( y'=\dfrac\pi2\cdot\dfrac{2}{\pi}=1 \).

17 A long product and quotient ★★★

\( \ln y=3\ln(x^2+1)+4\ln(2x-1)-2\ln(x+5) \), so \( \dfrac{y'}{y}=\dfrac{6x}{x^2+1}+\dfrac{8}{2x-1}-\dfrac{2}{x+5} \).

At \( x=1 \): \( y=\dfrac{8\cdot 1}{36}=\dfrac29 \) and \( \dfrac{y'}{y}=3+8-\dfrac13=\dfrac{32}{3} \).

Therefore \( y'(1)=\dfrac29\cdot\dfrac{32}{3}=\dfrac{64}{27} \).

18 Three layers ★★★

Layers: \( \sqrt{u} \), \( u=\sin w \), \( w=x^2 \). So \( y'=\dfrac{1}{2\sqrt{\sin(x^2)}}\cdot\cos(x^2)\cdot 2x=\dfrac{x\cos(x^2)}{\sqrt{\sin(x^2)}} \).

At \( x=\sqrt{\pi/6} \): \( x^2=\tfrac\pi6 \), \( \sin=\tfrac12 \), \( \cos=\tfrac{\sqrt3}{2} \). Then \( y'=\sqrt{\tfrac\pi6}\cdot\tfrac{\sqrt3}{2}\cdot\sqrt2=\sqrt{\tfrac\pi6}\cdot\tfrac{\sqrt6}{2}=\dfrac{\sqrt\pi}{2}\approx 0.89 \).

19 Second derivative of a circle ★★★

\( y''=-\dfrac{1\cdot y-x\,y'}{y^2}=-\dfrac{y+x^2/y}{y^2}=-\dfrac{y^2+x^2}{y^3}=-\dfrac{25}{y^3} \).

At \( (3,4) \): \( y''=-\dfrac{25}{64} \). Since \( y''\lt 0 \), the circle bends downward there, as expected on the upper half.

20 Inflating balloon ★★★

Both \( V \) and \( r \) depend on time \( t \). Chain rule: \( \dfrac{dV}{dt}=4\pi r^2\dfrac{dr}{dt} \).

With \( r=6 \) and \( \dfrac{dr}{dt}=0.5 \): \( \dfrac{dV}{dt}=4\pi(36)(0.5)=72\pi\approx 226.2 \) cubic inches per second (about 3,707 cm\( ^3 \)/s).

21 Proving the tangent rule ★★★

\( (\tan x)'=\dfrac{\cos x\cdot\cos x-\sin x\cdot(-\sin x)}{\cos^2x}=\dfrac{\cos^2x+\sin^2x}{\cos^2x}. \)

Since \( \cos^2x+\sin^2x=1 \), this equals \( \dfrac{1}{\cos^2x}=\sec^2x \).

22 Product of three functions ★★★

For three factors, \( (uvw)'=u'vw+uv'w+uvw' \). Here \( y'=e^x\sin x+x\,e^x\sin x+x\,e^x\cos x=e^x(\sin x+x\sin x+x\cos x) \).

At \( x=\tfrac\pi2 \): \( y'=e^{\pi/2}\left(1+\tfrac\pi2+0\right)=e^{\pi/2}\left(1+\tfrac\pi2\right)\approx 12.37 \).