Differentiation questions: Overview, Questions, Preparation

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Differentiation Differentiation Formulas Differentiation Rules Differentiation questions Continuity and Differentiabilit...

Rachit Kumar SaxenaManager-Editorial

Updated on Aug 27, 2021 12:56 IST

The differentiation of a capacity f(x) is spoken to as f'(x). In the event that f(x) = y, at that point f'(x) = dy/dx, which implies y is separated as for x. Before we begin tackling a few inquiries dependent on differentiation, let us see the overall differentiation equations utilized here.

Function f(x) = y	Differentiation of function f’(x) = dy/dx
xⁿ	nx^n-1
e^x	e^x
ln(x)	1/x
sin x	cos x
cos x	-sin x
tan x	sec²x
K (constant)	0

Table of Contents

What are Differential Questions?
Weightage of Differential Questions
Illustrated Examples on Differential Questions
FAQs on Differential Questions

What are Differential Questions?

1. Differentiate y = (tan x - 1)/ sec x
y = (tan x - 1)/ sec x
u = tan x - 1 ===> u' = sec² x - 0
v = sec x ===> v' = sec x tan x

2.Differentiate y = sin x / x²

y = sin x / x²

u = sin x ==> u' = cos x

u = x2 ==> v' = 2 x

dy/dx = [x² (cos x) - sin x(2x)]/(x²)²

dy/dx = [x² (cos x) - 2x sin x]/x⁴

dy/dx = x [x cos x - 2 sin x]/x⁴

dy/dx = [x cos x - 2 sin x]/x³

3. Differentiate y = tan θ (sin θ + cos θ)
y = tan θ (sin θ + cos θ)
u = tan θ ==> u' = sec² θ
v = sin θ + cos θ ==> v' = cos θ - sin θ
dy/dx = tan θ (cos θ - sin θ) + (sin θ + cos θ) (sec²θ)
= tan θ cos θ - tan θ sin θ + sin θ sec²θ + cos θ sec²θ

4. Differentiate y = cosec x ⋅ cot x
y = cosec x ⋅ cot x
u = cosec x ==> u' = -cosec x cot x
v = cot x ==> v' = -cosec² x
dy/dx = cosec x (-cosec² x) + cot x(-cosec x cot x)
= - cosec³ x - cosec x cot² x
= - (1/sin³ x) - (1/sin x) (cos² x/sin² x)
= - (1/sin³ x) - (cos² x/sin³ x)
dy/dx = - (1 + cos²x) /sin³ x

5. Differentiate 10x² with respect to x.
y = 10x²
y’ = d(10x²)/dx
y’ = 2.10.x = 20x
Therefore, d(10x2)/dx = 20 x

6. Compute the derivative of f(x) = sin²x.
f(x) = sin²x = sin x sin x
= d(sin x)/dx. sin x + sin x.d(sin x)/dx
= cos x. sin x + sin x cos x
= 2 sin x cos x

7. Differentiate tan²x.
Say, y = tan²x
dy/dx = d(tan²x)/dx
= 2tan^2-1x. d(tan x)/dx
= 2tan x sec²x
d[tan²x]/dx = 2tan x sec²x

Weightage of Differential Questions

In Class 11: The topic has been discussed with the basic explanations and the ways to solve the particular question. It has a weightage of 30 Marks.

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Illustrated Examples on Differential Questions

1. Differentiate sin(3x+5)

Solution.
Say, y = sin (3x+5)
dy/dx = d[sin(3x+5)]/dx
= cos (3x+5) d(3x+5)/dx [ By chain rule]
= cos (3x+5) [3]
y’ = 3 cos (3x+5)
d[sin(3x+5)]/dx = 3 cos (3x+5)

2.Differentiate 20x-4 + 9.

Solution.
y = 20x-4 + 9
y’ = d(20x-4 + 9)/dx
y’ = d(20x-4)/dx + d(9)/dx
y’ = -4.20.x-4-1+0
y’ = -80x-5
Therefore, d(20x-4 + 9)/dx = -80x-5
3. Differentiate x⁵ with respect to x.

Solution.

Given, y = x⁵
On differentiating w.r.t we get;
dy/dx = d(x⁵)/dx
y’ = 5x5-1 = 5x4
Therefore, d(x5)/dx = 5x4=20

FAQs on Differential Questions

Q: What do you mean by differentiation?

A: Differentiation permits us to discover paces of progress. For instance, it permits us to discover the pace of progress of speed as for time (which is quickening). It additionally permits us to discover the pace of progress of x as for y, which on a chart of y against x is the angle of the bend.

Q: Differentiate ln(10).

A: Since ln(10) is a constant, the derivative of ln(10) with respect to the variable “x” is 0.

Q: How can we differentiate log?

A: You can really utilize the subordinate of ln ⁡ ( x ) \ln(x) ln(x)natural log, left bracket, x, right enclosure (alongside the consistent various principle) to get the overall subsidiary of log ⁡ b ( x ) \log_b(x) logb(x)log, start base, b, end base, left bracket, x, right bracket.

Q: What is an equation to solve?

A: The equation is a mathematical expression containing the same symbol as. Equations often contain algebra. Algebra is used in mathematics when you don't know the right number in the calculation

Q: What are the types of equations?

A; There are three main forms of linear equations: point-slope shapes, standard forms, and slope-intercept forms