Class 12 Mathematics DERIVATIVES Notes

Class 12 Mathematics DERIVATIVES Notes
Derivatives Notes | Nepal eNotes

Unit 15

Calculus

Class 12 Mathematics

Derivatives

Class 12 Mathematics – Derivatives Notes PDF

On mobile, swipe inside the PDF to read all pages and pinch to zoom.

Contents / Quick Navigation

1. Derivative Recap2. Hyperbolic Functions 3. Derivatives of Hyperbolic Functions4. Inverse Hyperbolic Derivatives 5. L’Hospital’s Rule6. Tangent Using Derivatives 7. Normal Using Derivatives8. Derivative as Rate of Measure 9. Worked Examples10. Problem-Solving Strategy 11. Common Mistakes12. Important Exam Questions 13. One-Minute Revision

NEB / CDC Focus

This Grade 12 calculus chapter is organized around derivatives of hyperbolic and inverse-hyperbolic functions, L’Hospital’s rule for the indeterminate forms 0/0 and ∞/∞, tangent and normal using derivatives, and derivative as a rate of measure. Basic differentiation rules are included as prerequisite revision.

1. Derivative Recap

Derivative

For y=f(x), the derivative at x measures the instantaneous rate of change of y with respect to x and the slope of the tangent to the curve.

f′(x)=limh→0[f(x+h)−f(x)]/h.

1.1 Essential Rules

RuleFormula
Powerd(xⁿ)/dx=nxⁿ⁻¹
Sum(u+v)′=u′+v′
Product(uv)′=u′v+uv′
Quotient(u/v)′=(vu′−uv′)/v²
Chaind[f(g(x))]/dx=f′(g(x))g′(x)
Derivative as Tangent Slope P(a,f(a))tangent Slope of tangent at x=a is f′(a).
Figure 1: Geometric meaning of a derivative.

2. Hyperbolic Functions

Hyperbolic functions are defined using exponential functions:

sinh x=(eˣ−e⁻ˣ)/2 cosh x=(eˣ+e⁻ˣ)/2 tanh x=sinh x/cosh x

Important identity:

cosh²x−sinh²x=1.
sinh x and cosh x sinh xcosh x sinh is odd; cosh is even and cosh x≥1.
Figure 2: Characteristic shapes of the main hyperbolic functions.

3. Derivatives of Hyperbolic Functions

d(sinh x)/dx=cosh x d(cosh x)/dx=sinh x d(tanh x)/dx=sech²x d(coth x)/dx=−csch²x d(sech x)/dx=−sech x tanh x d(csch x)/dx=−csch x coth x

3.1 Derivation of d(sinh x)/dx

Using sinh x=(eˣ−e⁻ˣ)/2,

d(sinh x)/dx=(eˣ+e⁻ˣ)/2=cosh x.

3.2 Chain Rule

d[sinh u]/dx=cosh u·du/dxd[tanh u]/dx=sech²u·du/dx.
Worked Example

Differentiate y=sinh(3x²).

dy/dx=cosh(3x²)·6x=6x cosh(3x²).

Hyperbolic Differentiation Pair sinh xderivative → cosh x cosh xderivative → sinh x
Figure 3: Unlike sine and cosine, differentiating cosh does not introduce a minus sign.

4. Derivatives of Inverse Hyperbolic Functions

FunctionDerivativeTypical real-domain note
sinh⁻¹x1/√(1+x²)all real x
cosh⁻¹x1/√(x²−1)x>1 for ordinary derivative
tanh⁻¹x1/(1−x²)|x|<1

4.1 Derivative of sinh⁻¹x

Let y=sinh⁻¹x. Then x=sinh y.

dx/dy=cosh y.

Since cosh²y−sinh²y=1,

cosh y=√(1+x²).

Hence

dy/dx=1/√(1+x²).
Notation

sinh⁻¹x denotes the inverse hyperbolic sine, not 1/sinh x. The reciprocal of sinh is csch.

5. L’Hospital’s Rule

L’Hospital’s rule is used for limits that produce the indeterminate forms 0/0 or ∞/∞.

L’Hospital’s Rule
If lim f(x)/g(x) is 0/0 or ∞/∞, then under the usual differentiability conditions,lim f(x)/g(x)=lim f′(x)/g′(x),

provided the latter limit exists (finite or infinite) in the required sense.

  1. Substitute first and verify the form is 0/0 or ∞/∞.
  2. Differentiate numerator and denominator separately.
  3. Evaluate the new limit.
  4. If the new form is again 0/0 or ∞/∞, the rule may be applied again.
Worked Example

Evaluate limx→0(eˣ−1)/x.

Substitution gives 0/0.

lim eˣ/1=1.

Worked Example

Evaluate limx→∞(2x²+1)/(5x²−3) by L’Hospital.

Form is ∞/∞.

lim 4x/10x=2/5.

L’Hospital Decision Flow Substitute into limitWhat form? 0/0 or ∞/∞?Differentiate top/bottom Evaluate againobtain limit
Figure 4: Verify the indeterminate form before using L’Hospital’s rule.
Common Misuse

Do not use L’Hospital directly for a determinate form such as 0/5 or for forms like 0·∞ until they are first transformed into a suitable quotient form.

6. Tangent to a Curve Using Derivatives

For y=f(x), at the point (a,f(a)), tangent slope is

m_t=f′(a).

Therefore the tangent equation is

y−f(a)=f′(a)(x−a).
Worked Example

Find tangent to y=x² at (2,4).

dy/dx=2x ⇒ m=4.

y−4=4(x−2) ⇒ y=4x−4.

Tangent from the Derivative Pslope=f′(a)
Figure 5: The derivative supplies the tangent slope directly.

7. Normal to a Curve Using Derivatives

The normal is perpendicular to the tangent. If f′(a)≠0,

m_n=−1/f′(a).

Thus

y−f(a)=−[1/f′(a)](x−a).
Horizontal Tangent Case

If f′(a)=0, the tangent is horizontal and the normal is vertical: x=a.

Worked Example

For y=x² at (2,4), tangent slope is 4, so normal slope is −1/4.

y−4=−(1/4)(x−2).

Tangent and Normal tangentnormalP
Figure 6: Tangent and normal are perpendicular at the point of contact.

8. Derivative as a Rate of Measure

If a quantity y depends on time t, then dy/dt is its instantaneous rate of change.

QuantityDerivative interpretation
Position s(t)velocity v=ds/dt
Velocity v(t)acceleration a=dv/dt=d²s/dt²
Area A(t)dA/dt = rate of change of area
Volume V(t)dV/dt = rate of change of volume

8.1 Related Rates

When several quantities change with time and are connected by an equation, differentiate the entire relation with respect to time.

Worked Example: Expanding Circle

A circle’s radius increases at dr/dt=2 cm/s. Find dA/dt when r=5 cm.

A=πr².

dA/dt=2πr·dr/dt=2π(5)(2)=20π cm²/s.

Worked Example: Motion

If s=t³−3t²+2t, find velocity at t=2.

v=ds/dt=3t²−6t+2.

v(2)=12−12+2=2.

Related Rates Chain Geometric relationA=πr² Differentiate in tdA/dt=2πr dr/dt Substitute instantfind required rate
Figure 7: Related-rate problems differentiate a relationship before substituting the instant.

9. Worked Examples

Example 1: Hyperbolic Derivative

Differentiate y=cosh(2x).

dy/dx=2sinh(2x).

Example 2: tanh

Differentiate y=tanh(x²).

dy/dx=2x sech²(x²).

Example 3: Inverse Hyperbolic

Differentiate y=sinh⁻¹(3x).

dy/dx=3/√(1+9x²).

Example 4: L’Hospital 0/0

Evaluate limx→0 sinh x/x.

0/0 ⇒ lim cosh x=1.

Example 5: Tangent and Normal

For y=x³ at (1,1), f′(x)=3x², so tangent slope is 3 and normal slope is −1/3.

Tangent: y−1=3(x−1).

Normal: y−1=−(1/3)(x−1).

Example 6: Volume Rate

For a sphere V=(4/3)πr³ with dr/dt=1,

dV/dt=4πr²dr/dt=4πr².

10. Problem-Solving Strategy

  1. Identify the function type before choosing a differentiation rule.
  2. For composite hyperbolic functions, include the chain-rule factor.
  3. For inverse hyperbolic functions, note the real-domain restrictions when relevant.
  4. Before L’Hospital, substitute and verify 0/0 or ∞/∞.
  5. For tangent/normal, first find dy/dx, then evaluate it at the given point.
  6. For a normal, use the negative reciprocal unless the tangent is horizontal/vertical.
  7. For rate problems, write a relation among the changing quantities, differentiate with respect to time, then substitute the instant.
  8. Always attach appropriate units to physical rates.

11. Common Mistakes

cosh derivative sign

d(cosh x)/dx=sinh x, not negative sinh.

Chain rule omitted

For sinh u, multiply by du/dx.

Inverse vs reciprocal

sinh⁻¹x is inverse hyperbolic sine, not csch x.

L’Hospital misuse

Verify the indeterminate form before differentiating numerator and denominator.

Normal slope

Normal slope is the negative reciprocal of tangent slope, not simply negative.

Substituting too early

In related rates, differentiate the relation first, then substitute the numerical instant.

12. Formula Sheet

(sinh x)′=cosh x(cosh x)′=sinh x (tanh x)′=sech²x(sinh⁻¹x)′=1/√(1+x²) (cosh⁻¹x)′=1/√(x²−1)(tanh⁻¹x)′=1/(1−x²) L’Hospital for 0/0 or ∞/∞: lim f/g = lim f′/g′ Tangent: y−f(a)=f′(a)(x−a) Normal: y−f(a)=−[1/f′(a)](x−a) Rate: dy/dt = derivative of y with respect to time

13. Important Exam Questions

Short

  1. Define derivative geometrically and as a rate of change.
  2. State derivatives of sinh, cosh and tanh.
  3. State derivatives of inverse sinh, cosh and tanh.
  4. State L’Hospital’s rule and the indeterminate forms covered.
  5. Write tangent and normal equations at a point.

Long / Numerical

  1. Differentiate composite hyperbolic functions.
  2. Derive the derivative of sinh⁻¹x.
  3. Evaluate limits of forms 0/0 or ∞/∞ using L’Hospital’s rule.
  4. Find tangent and normal to a curve at a specified point.
  5. Solve velocity/acceleration or geometric related-rate problems.

14. One-Minute Revision

  • A derivative is an instantaneous rate of change and tangent slope.
  • (sinh x)′=cosh x.
  • (cosh x)′=sinh x.
  • (tanh x)′=sech²x.
  • (sinh⁻¹x)′=1/√(1+x²).
  • L’Hospital applies directly to 0/0 and ∞/∞.
  • Tangent slope at x=a is f′(a).
  • Normal slope is −1/f′(a) when finite and non-zero.
  • Velocity is ds/dt; acceleration is d²s/dt².
  • In related rates, differentiate before substituting the instant.

15. Diagram Practice

  • Derivative as tangent slope.
  • Graphs of sinh and cosh.
  • Hyperbolic differentiation pair.
  • L’Hospital decision flow.
  • Tangent to a curve.
  • Tangent and normal together.
  • Related-rates chain.

Also Visit

Class 12 Mathematics Notes Class 12 Notes Practical File Question Papers Important Questions
Source handling: The original Nepal eNotes PDF remains embedded above. The typed section follows the verified NEB/CDC syllabus and is designed as a searchable, responsive study companion. Where the PDF viewer does not expose handwritten page text, the typed section is a syllabus-aligned reconstruction and is not claimed to be a word-for-word transcription.

Original Nepal eNotes page: Derivatives Notes.
KEEP LEARNING View All Posts

Discussion

Share a helpful question, idea, or explanation with other students.

Leave a Comment

Write a clear question, answer, or helpful explanation.
Your email will not be published.

Download Our Offline App

Study class-wise notes even when internet is not available. Get the app from Play Store.

Nepal eNotes offline app preview
Get it on Google Play