Calculus: Differentiation

Question 1

What is the differential of y = axⁿ?

Answer 1

if y = axⁿ, then dy/dx = anx^n-1

also

if f(x) = axⁿ, then f ‘(x) = anx^n-1

Question 2

Differentiate y = x³.sinx with respect to x.

Answer 2

If y = x³·sinx

dy/dx = x³(cos x) + sinx(3x²)

= x³cosx + 3x²sinx

= x²(xcosx + 3sinx)

Question 3

What is the differential of y = ax?

Answer 3

If y = ax, then dy/dx = a

also

if f(x) = ax, then f ‘(x) = a

Question 4

What is the differential of y = a?

Answer 4

If y = a, then dy/dx = 0

also

if f(x) = a, then f ‘(x) = 0

Question 5

What is the second derivative of y = axⁿ

Answer 5

If y = axⁿ, then dy/dx = anx^n-1

and

d²y/dx² = (n-1)anx^n-2

e.g.

if y = 2x³, then dy/dx = 6x²

and

d²y/dx² = 12x

Note: y = f(x)

dy/dx = f ‘(x)

d²y/dx² = f ‘‘(x)

Question 6

Given the equation y = f(x) and a point, P, on the curve with co-ordinates (x₁, y₁), find the equation of the tangent to the curve.

Answer 6

The equation of a line can be shown to be:

(y - y₁) = m(x - x₁)

where m = gradient of the line.

The gradient is found using differentiation.

m = dy/dx

Question 7

Given the equation y = f(x) and a point, P, on the curve with co-ordinates (x1, y1), find the equation of the normal to the curve.

Answer 7

The equation of a line normal to the tangent can be shown to be:

(y - y1) = (-1/m)(x - x1)

where m = gradient of the line tangent to the curve.

This gradient is found using differentiation.

m = dy/dx

Question 8

What is the derivative of Sin x?

Answer 8

If y = Sin x, then dy/dx = cos x

Question 9

What is the derivative of Cos x?

Answer 9

if y = Cos x, then dy/dx = - Sin x

Question 10

What is the derivative of e^x with respect to x?

Answer 10

If y = e^x, then dy/dx = e^x

Question 11

Differentiate y = uv , where u and v are functions of x.

Hint: Product rule.

Answer 11

If y = uv , then dy/dx = u(dv/dx) + v(du/dx)

i.e. To differentiate a product of two functions:

Put down the first (differentiate the second) + put down the second (differentate the first)

Question 12

Differentiate y = x⁴.cos x with respect to x.

Answer 12

If y = x⁴ cos x , then dy/dx = x⁴(-sin x) + cosx(4x³)

= 4x³cosx - x⁴sinx

=x³(4cosx - xsinx)

Question 13

Differentiate y = x⁵.e^x with respect to x.

Answer 13

If y = x⁵ e^x, then dy/dx = x⁵(e^x) + e^x(5x⁴)

= x⁵e^x +5x⁴e^x

=x⁴e^x(x +5)

Question 14

Differentiate y = e^x.sinx with respect to x

Answer 14

If y = e^x.sinx, then dy/dx = e^x(cosx) + sinx(e^x)

= e^xcosx + e^xsinx

=e^x(cosx + sinx)

Question 15

Differentiate y = 4x³.sinx with respect to x

Answer 15

If y = 4x³.sinx, then dy/dx = 4x³(cosx) + sinx(12x²)

= 4x³cosx + 12x²sinx

= 4x²(xcosx + 3sinx)

Question 16

Differentiate y = e^x.cosx with respect to x

Answer 16

If y = e^x.cosx, then dy/dx = e^x(-sinx) + cosx(e^x)

= e^xcosx - e^xsinx

= e^x(cosx - sinx)

Question 17

Differentiate y = cosx.sinx with respect to x

Answer 17

If y = cosx.sinx, then dy/dx = cosx(cosx) + sinx(-sinx)

= cos²x - sin²x

= cos(2x)

Question 18

Differentiate y = 3x³.e^x with respect to x

Answer 18

If y = 3x³.e^x, then dy/dx = 3x³(e^x) + e^x(9x²)

= 3x³e^x + 9x²e^x

= 3x²e^x(x + 3)

Question 19

Differentiate y = 2x⁵.cosx with respect to x

Answer 19

If y = 2x⁵.cosx, then dy/dx = 2x⁵(-sinx) + cosx(10x⁴)

= 10x⁴cosx - 2x⁵sinx

=2x⁴(5cosx - xsinx)

Question 20

Differentiate y = u/v , where u and v are functions of x.

Hint: Quotient rule.

Answer 20

To differentiate a quotient of two functions:

[Put down the bottom (differentiate the top) - put down the top (differentiate the bottom)] all over the bottom squared.

“low d high over high d low, all over the square of what’s below”

Question 21

What is the derivative of tan x?

Hint: Quotient Rule

Answer 21

Tan x = sin x / cos x

If y = sinx / cosx,

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

= (cos²x + sin²x) / cos²x

= 1 / cos²x

= sec²x

Question 22

Differentiate y = sin x / x² with respect to x.

Answer 22

If y = sin x / x², then dy/dx = [x²(cosx) - sinx (2x)] / (x²)²

= [x(xcos x - 2sin x)] / x⁴

= (xcos x - 2sin x) / x³

Question 23

Differentiate y = 5e^x / cos x with respect to x.

Answer 23

If y = 5e^x / cos x,

then dy/dx = [cos x.(5e^x) - 5e^x.(-sinx)] / (cos x)²

= [5e^x.(cos x + sin x)] / cos² x

Question 24

Differentiate y = cos x / sin x with respect to x.

Answer 24

If y = cos x / sin x,

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

= (-sin² x - cos² x) / sin² x

= - (sin² x + cos² x) / sin² x

= - 1 / sin² x

= - cosec² x

Question 25

Differentiate y = 3x²/cos x with respect to x

Answer 25

If y = 3x²/cos x,

then dy/dx = [cos x.(6x) - 3x².(-sin x)] / (cos x)²

= [3x (2 cos x + x sin x)] / cos² x

Question 26

Differentiate y = tan x / e^x with respect to x

Answer 26

If y = tan x / e^x

then dy/dx = [e^x (sec² x) - tan x (e^x)] / (e^x)²

= [e^x (sec² x - tan x)] / (e^x)²

= (sec² x - tan x) / e^x

Question 27

Differentiate f (x) = g (h (x)) with respect to x

Answer 27

If f (x) = g (h (x)),

Then f ‘(x) = (g (h (x)))’

= g ‘(h (x)) h’ (x)

or

dy/dx = dy/du x du/dx