Techniques of differentiation

Identify the outer and inner operations

First identify the structure: sum, product, quotient or composition. Choose the rule from that structure, then differentiate each required factor. A composition also needs the derivative of its inner function.

The basic rules are $d(a)/dx=0$, $d(x^n)/dx=nx^{n-1}$, $d(af)/dx=af^{\prime}$ and $d(f\pm g)/dx=f^{\prime}\pm g^{\prime}$, on intervals where the real functions are differentiable.

A general real exponent does not make $x^n$ real and differentiable at every real x.

For $y=f(u)$ and $u=g(x)$, $dy/dx=(dy/du)(du/dx)$.

Work from the outermost operation inward.

For products use $(uv)^{\prime}=uv^{\prime}+vu^{\prime}$.

For quotients use $(u/v)^{\prime}=(vu^{\prime}-uv^{\prime})/v^2$, with $v\ne0$.

When simplifying a product-rule answer, the source recommends factoring the smaller power; for its rational example it recommends factoring $2x^2$ before expansion.

Trigonometric derivatives use radians

Identify the inner function before choosing a derivative. Use radian input, keep the inner derivative as a factor, then check where the original function is defined.

For a differentiable inner function u, all six source rows are:

Function: $\sin u$

• Derivative with respect to x: $\cos u\,u^{\prime}$

Function: $\cos u$

• Derivative with respect to x: $-\sin u\,u^{\prime}$

Function: $\tan u$

• Derivative with respect to x: $\sec^2u\,u^{\prime}$

Function: $\sec u$

• Derivative with respect to x: $\sec u\tan u\,u^{\prime}$

Function: $\csc u$

• Derivative with respect to x: $-\csc u\cot u\,u^{\prime}$

Function: $\cot u$

• Derivative with respect to x: $-\csc^2u\,u^{\prime}$

Setting u=x gives the source’s basic column.

The source spells csc as cosec and shades the basic secant/cosecant rows as listed in MF27.

Preserve the relevant nonzero sine/cosine denominators.

Degrees would introduce an additional angle-conversion factor; these displayed formulas assume radians.

Exponentials and logarithms

The four source rows, including their general inner-function forms, are:

Function: $e^u$

• Derivative with respect to x: $e^u u^{\prime}$

Function: $a^u$

• Derivative with respect to x: $a^u\ln a\,u^{\prime}$

Function: $\ln u$

• Derivative with respect to x: $u^{\prime}/u$

Function: $\log_a u=\ln u/\ln a$

• Derivative with respect to x: $u^{\prime}/(u\ln a)$

The exponential base is positive.

A real logarithm additionally needs u>0 and base a>0, a≠1.

Setting u=x gives every printed basic formula.

Simplifying an answer does not change its original domain: for example, the identity used in Example 7(b) is valid only where its tangent/secant expression exists.

Implicit differentiation follows a local branch

Treat the dependent variable as a function throughout the calculation. Differentiate both sides, collect its derivative terms, and check their coefficient before dividing.

An explicit expression gives y entirely in terms of x, as in $y=2/x+3x^2$ or $y=4x/(x-1)$.

The source contrasts these with $y=2/x+3y^2$, $xy=y^2-10$ and $\sqrt{x+1}=y^x+e^y$.

An implicit relation need not give one global function.

On a differentiable local branch $y=y(x)$, the chain rule is $d[g(y)]/dx=(dg/dy)(dy/dx)$.

Collect all derivative terms before dividing, and check whether their coefficient is zero.

Examples 11 and Self-Practice 4 apply this procedure; the Teacher supplies the missing product and chain steps in Example 11(b).

Logarithms separate variable powers

For $u^v$ with variable base and exponent, or complicated products and quotients, the source procedure is: take natural logarithms on both sides, simplify logarithms where possible, then differentiate implicitly with respect to x.

Real logarithms require positive arguments.

Example 12 keeps its printed condition $x,y>0$, and Self-Practice 5 proves the derivative of $x^x$ on $x>0$.

A quotient obtained by solving for $dy/dx$ applies only where its denominator is nonzero; retain the preceding undivided equation at exceptional points.

Inverse trigonometric functions use principal ranges

Read the function domain and principal range before differentiating. Then check the stricter conditions needed by the displayed derivative formula; endpoint composites require their own analysis.

Here $\sin^{-1}$, $\cos^{-1}$ and $\tan^{-1}$ mean arcsine, arccosine and arctangent. They are not $1/\sin$, $1/\cos$ or $1/\tan$; the source explicitly distinguishes inverse functions from reciprocals.

Inverse function: $\sin^{-1}x$

• Real domain: $[-1,1]$
• Principal range: $[-\pi/2,\pi/2]$

Inverse function: $\cos^{-1}x$

• Real domain: $[-1,1]$
• Principal range: $[0,\pi]$

Inverse function: $\tan^{-1}x$

• Real domain: $\mathbb R$
• Principal range: $(-\pi/2,\pi/2)$

For a differentiable inner function u, the three source basic/general rows are:

Function: $\sin^{-1}u$

• Basic derivative (u=x): $1/\sqrt{1-x^2}$
• General derivative: $u^{\prime}/\sqrt{1-u^2}$

Function: $\cos^{-1}u$

• Basic derivative (u=x): $-1/\sqrt{1-x^2}$
• General derivative: $-u^{\prime}/\sqrt{1-u^2}$

Function: $\tan^{-1}u$

• Basic derivative (u=x): $1/(1+x^2)$
• General derivative: $u^{\prime}/(1+u^2)$

The source shades the three basic formulas as being in MF27. It prints $-1\le x\le1$ (and $-1\le u\le1$) beside the first two derivative rules. Those are the function domains: the displayed derivative quotients require $|x|<1$ or $|u|<1$ so their denominators are nonzero.

Endpoint composites need separate analysis; a constant inner function can, for example, give a constant endpoint value with derivative zero.

The arctangent rule has no corresponding real endpoint restriction.

Why arcsine uses the positive square root

The source proof sets $y=\sin^{-1}x$, so $\sin y=x$.

Implicit differentiation gives $\cos y\,y^{\prime}=1$ and $y^{\prime}=1/\cos y$.

On the principal range $[-\pi/2,\pi/2]$, $\cos y\ge0$, hence $\cos y=\sqrt{1-\sin^2y}=\sqrt{1-x^2}$.

For $-1<x<1$, cosine is strictly positive, so division gives $y^{\prime}=1/\sqrt{1-x^2}$.

The endpoints are excluded from this finite derivative formula.

Both copies point to Appendix page 20, Examples D and E, for the arccosine and arctangent proofs; both proofs are now included as Appendix Examples D and E below.

A calculator evaluates a derivative at a point

Before using the calculator, identify the function, variable and evaluation point. After calculating, check whether the question needs a numerical value or an exact symbolic result.

Section 4 is marked self-reading. Example 14 retains both printed menu routes to nDeriv, the function/variable/evaluation-point template, ENTER and the results 4 and 0.600 (three significant figures).

Use a numerical derivative as a value at the specified point; it is not a replacement for an exact symbolic proof.

Appendix Examples A–E are included below. The sine/exponential numerical limits are identified as standard limits motivated by source calculator observations, rather than proved by finite samples.

The Teacher-only syllabus note is integrated into the sine proof; both calculator workflows distinguish missing values from rounded near-zero displays.

The full page 21 mastery checklist and study map are below.

The closing Discussion and Review questions reuse the existing reviewed chapter sets, preserving the printed source key and its recorded qualifications.