First Principles and the Rules: Quick Revision

A brief note on First Principles and the Rules. Read it before a test. Try each question at the end before you open its answer.

The definition

• Derivative at a point: $f'(a) = \displaystyle\lim_{h \to 0} \dfrac{f(a + h) - f(a)}{h}$, the slope of the tangent at $x = a$.
• Secant: the line through $(a, f(a))$ and $(a + h, f(a + h))$. Its slope is the quotient above.
• Tangent line: $y - f(a) = f'(a)(x - a)$.
• Notation: $f'(x)$, $\dfrac{dy}{dx}$. The UK says gradient and the US says slope.
• Corner: a corner such as $|x|$ at $0$ has no derivative.

The rules

• Constant: $(c)' = 0$. Power: $(x^n)' = nx^{n-1}$. Multiple: $(cf)' = cf'$. Sum: $(f + g)' = f' + g'$.
• Product: $(uv)' = u'v + uv'$.
• Quotient: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$.

Standard derivatives, with $x$ in radians

• Sine and cosine: $(\sin x)' = \cos x$ and $(\cos x)' = -\sin x$.
• Tangent and friends: $(\tan x)' = \sec^2 x$, $(\sec x)' = \sec x\tan x$, $(\csc x)' = -\csc x\cot x$, $(\cot x)' = -\csc^2 x$.
• Exponential and logarithm: $(e^x)' = e^x$, $(\ln x)' = \dfrac{1}{x}$, $(a^x)' = a^x\ln a$.

A rate

• Meaning: $\dfrac{dy}{dx}$ is the change of $y$ per unit of $x$, with the unit of $y$ per unit of $x$.
• Speed: $v = \dfrac{ds}{dt}$ in m/s.
• Circle: $\dfrac{dA}{dr} = 2\pi r$.
• Class 12: the chain rule, implicit and logarithmic differentiation come in the chapter on continuity and differentiability.

A worked example

Differentiate $f(x) = x^2\cos x$ and find $f'(\pi)$.

1. Take $u = x^2$ and $v = \cos x$. Then $u' = 2x$ and $v' = -\sin x$.
2. Product rule. $f'(x) = 2x\cos x - x^2\sin x$.
3. At $x = \pi$. $2\pi \times (-1) - \pi^2 \times 0 = -2\pi$.

Practice with new numbers

A program to try

The program compares each derivative with a small-step difference quotient.

Mistakes to avoid

• Product of the derivatives. Use $u'v + uv'$.
• The top of the quotient rule. It is $u'v - uv'$, and the bottom is $v^2$.
• Degrees. The trigonometric derivatives need radians.
• Lost factor. $(x^5)' = 5x^4$.
• A rate without a unit. Write the unit of $y$ per unit of $x$.

Check yourself

1. Find the derivative of $f(x) = x^3$ at $x = 2$ from the definition.

2. Find the slope of $y = \cos x$ at $x = \dfrac{\pi}{6}$.

3. Differentiate $y = x\sin x$.

4. Differentiate $y = \dfrac{x - 1}{x + 1}$.

5. Find the tangent to $y = x^2$ at $x = 3$.

Sources

• OpenStax, Calculus Volume 1, Chapter 3 on derivatives.
• NCERT, Mathematics Textbook for Class XI, Chapter 12.