## The Calculus Primer (2011)

### Part VII. Differentials

### Chapter 27. SUMMARY OF DIFFERENTIAL NOTATION

*dc =* 0

*d*(*cv*) = *c dv*

*d*(*u* + *v*) = *du* + *dv*

*d*(*uv*) = *u dv* + *v du*

*d*(*v ^{n}*) =

*nv*

^{n−1}*dv*

*d*(*x ^{n}*) =

*nx*

^{n}^{−1}

*dx*

*d*(log* _{a} v*) = log

_{a}*e*

*d*(log *v*) =

*d*(*a ^{v}*) =

*a*log

^{v}*a*

*dv*

*d*(*e ^{v}*) =

*e*

^{v}dv*d*(*u ^{v}*) =

*vu*

^{v}^{−1}

*du*+ log

*u*·

*u*·

^{v}*dv*

*d*(sin *v*) = cos *v dv*

*d*(cos *v)* = − sin *v* *dv*

*d*(tan *v*) = sec^{2} *v* *dv*

*d*(cot *v*) = − csc^{2} *v* *dv*

*d*(sec *v*) = sec *v* tan *v* *dv*

*d*(csc *v*) = − csc *v* cot *v* *dv*

**7—6.Differentiation with Differentials.** This may be illustrated by the following examples.

EXAMPLE 1.Find the differential of

*y* = *x*^{5} *+* 3*x*^{2}.

*Solution. dy* = 5*x*^{4} *dx +* 6*x dx,*

or*dy* = (5*x*^{4} + 6*x*) *dx.*

EXAMPLE 2.Find the differential of

EXAMPLE 3.Find *dy* for *y* = log sin 2*x.*

*Solution.* *d*(log *v*) = ,

EXAMPLE 4.Find *dy* from

*a*^{2}*x*^{2} + *b*^{2}*y*^{2} = *a*^{2}*b*^{2}.

*Solution.* 2*a*^{2}*x dx* + 2*b*^{2}*y dy* = 0,

EXAMPLE 5.Find *dρ* from

*ρ*^{2} = *a*^{3} sin 3*θ*.

*Solution.* 2*ρ* *dρ* = *a*^{3} cos 3θ·3*dθ*,

**7—7.Successive Differentials.** Consider the function *y* = *f*(*x*). We may regard *d*(*dy*) as the *second differential* of *y,* or the second differential of the function; it is represented by the symbol *d*^{2}*y.* In the same way,

*d*[*d*(*dy*)],or*d*^{3}*y,*

is the *third differential of y*; and *d ^{n}y* is the

*nth differential of y.*

The reader must be careful in interpreting the symbolism used here. Thus

*d*^{2}*v* is the *second differential* of *v,* and equals *d*(*dv*).

*dv*^{2} is the *square* of *dv,* and may also be written (*dv*)^{2}.

*d*(*v*^{2}) is the *first* *differential* of *v*^{2}*,* and equals 2*v* *dv.*

In general, in the function *y* = *f*(*x*), the differential of the independent variable, namely, *dx,* is independent of *x,* and must be regarded as a constant when differentiating with respect to *x.* Consequently:

*dy = f′*(*x*)*dx,*

and, since *dx* is a constant,

*d*(*dy*) = *d*^{2}*y* = (*dx*)·[*f″*(*x*) *dx*] *= f″*(*x*)(*dx*)^{2}.

Similarly,*d ^{3}y* =

*f″′*(

*x*) (

*dx*)

^{3},

and*d ^{n}y = f*

^{(n)}(

*x*)(

*dx*)

^{n}.The reader should note that, since (*dx*)^{2} = *dx*^{2}, the differential notation

*d*^{2}*y=f″*(*x*)(*dx*)^{2}(1)

is equivalent to the familiar derivative notation

simply by dividing both sides of (1) by (*dx*)^{2}, or *dx*^{2}. Similarly,

**EXERCISE 7—1**

*Differentiate the following, using differentials:*

**1.** *y* = *x*^{3} + 5*x*^{2} − 3

**2.** *y* = (5*x* + 2)^{3}

**3.** *y* =

**4.** *y* = *ke ^{tx}*

**5.** *y = e ^{x}* log

*x*

**6.** *ρ* = sin2*θ* + 2cos *θ*

**7.** *f*(*x*) = (log *x*)^{4}

**8.** *ρ* = *a*^{2} sec 3*θ*

*Find dy for each of the following:*

**9.** *x*^{2} + *y*^{2} = *r*^{2}

**10.** *b*^{2}*x*^{2} − *a*^{2}*y*^{2} = *a*^{2}*b*^{2}

**11.** *xy = k*

*Find d*^{3}*y for each of the following:*

**12.** *y* = *x*^{4} − 5*x*^{3} + 4*x*^{2} − 7

**13.** *y = e ^{x}* sin

*x*

**14.** *y* =

**15.** *y* =