Differentiability and the Rules of Differentiation: Quick Revision

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

Differentiability

• Definition: $f'(a) = \displaystyle\lim_{h \to 0}\frac{f(a + h) - f(a)}{h}$, when the limit exists.
• Test: the left derivative equals the right derivative.
• Link: differentiable implies continuous. Continuous does not imply differentiable.
• Ways to fail: a corner, a vertical tangent, a break.

The rules

• Product: $(uv)' = u'v + uv'$. Quotient: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$.
• Chain rule: $\dfrac{dy}{dx} = \dfrac{dy}{du}\times\dfrac{du}{dx}$.
• Exponentials and logarithms: $(e^x)' = e^x$, $(a^x)' = a^x\ln a$, $(\ln x)' = \dfrac{1}{x}$.
• Inverse trigonometry: $(\sin^{-1}x)' = \dfrac{1}{\sqrt{1 - x^2}}$, $(\cos^{-1}x)' = -\dfrac{1}{\sqrt{1 - x^2}}$, $(\tan^{-1}x)' = \dfrac{1}{1 + x^2}$.
• Inverse functions: if $y = f^{-1}(x)$, then $\dfrac{dy}{dx} = \dfrac{1}{f'(y)}$.

Three methods

• Implicit: differentiate both sides, treat $y$ as a function of $x$, then solve for $\dfrac{dy}{dx}$.
• Logarithmic: take $\ln$ of both sides, differentiate, and multiply by $y$. For $x^x$ the result is $x^x(1 + \ln x)$.
• Parametric: $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$.

A worked example

Is $f(x) = x^2$ for $x \le 3$ and $f(x) = 6x - 9$ for $x > 3$ differentiable at $x = 3$?

1. Continuity. $f(3) = 9$ and the right piece gives $18 - 9 = 9$.
2. Left derivative. $2x = 6$ at $x = 3$.
3. Right derivative. $6$.
4. The answer. The two derivatives are equal, so $f$ is differentiable at $3$ and $f'(3) = 6$.

Practice with new numbers

A program to try

The program checks the chain rule and two standard derivatives against numerical slopes.

Mistakes to avoid

• Forgetting the inside derivative in the chain rule.
• Testing one side only when checking differentiability.
• Using the power rule on $a^x$. The derivative is $a^x\ln a$.
• Dropping $\dfrac{dy}{dx}$ when differentiating $y^2$ implicitly.
• Dividing the wrong way in parametric form.

Check yourself

1. Differentiate $\sin^{-1}(3x)$.

2. Differentiate $\ln(\cos x)$.

3. Find $\dfrac{dy}{dx}$ for $x = t^3$, $y = t^6 - 1$.

4. Is $f(x) = |x + 1|$ differentiable at $x = -1$?

5. Find $\dfrac{dy}{dx}$ for $x^2 + y^2 = 9$.

Sources

• OpenStax, Calculus Volume 1, Chapter 3.
• NCERT, Mathematics Part I for Class XII, Chapter 5.