Calculus Flashcards
What is a derivative
Derivative of a function = dy/dx = slope at a point
It gives us the instantaneous rate of change of a function.
Eg. If derivative of a function = 5x, then the instantaneous rate of change of that function when x=3, would be 15.
Why is the derivative of a constant value 0
For a constant function, say c, The value of y remains same as long as x stays constant say “c”
Which means f’(c) = 0
Derivative of a constant is always
0 ,because in the graph: when we check the slope at any point is 0.
In a graph in which y=x³ the slope at any point in the graph (derivative) :
3x³⁻¹ = 3x²
This rule is called the power rule.
When you take f(x) = C (a constant) the slope of it is
So it means that whatever value you give for x , the value of the y (fx) coordinate always has the same value 7. As there is no change in the y coordinate , the slope is 0.
The shape of the graph when y = f(x) = x²
Parabola
To find the indefinite integral of a f(x) means
Finding the anti derivative of f(x)
That would be another function.
Reverse power rule
∫x³ dx = (x ³⁺¹)/3+1 + C
An angle in xy plane, is said to be in its standard form, when
The vertex is at the origin and the initial ray lies also get the positive x axis.
Sin of 37
3/5
Sin of 53
4/5
Cos of 37
4/5
Tangent
Straight line touching a curve at a particular point.
If c is a constant d/dx of c =
0
Derivative of x³ is
3x³⁻¹ = 3x²
This rule is called the power rule.
d/dx (Cx) =
C d/dx (x)
This rule is called constant multiple rule
The derivative of the negative of a differentiable function is the negative of the function’s derivative
d/dx (-u) = -1 d/dx(u)
Derivative of the sum of 2 differentiable functions is the sum of their derivatives.
This rule is called
The sum rule:
d/dx(u-v) = d/dx(u) - d/dx(v)
In prime notation :
‘ (derivative)
″ (double derivative)
‴ (third derivative)
⁗ (fourth derivative)
Product rule
d/dx (uv) = u. d/dx (v) + v. d/dx (u)
Derivative of sin x
Cos x
Derivative of cos x is
- sin x.
Derivative of e raised to x
e raised to x
Derívate of the natural log of x =
1/x
Chain rule :
y = f(g(x)) Then f(x)/f(y) = f’(g(x)) . g’(x)
Chain rule is also known as
Outside inside rule
d/dx of (dy/dx) is written
d²y/dx²
At maxima
dx/dy = 0
d²x/dy² < 0
In minima d²x/dy² is
Positive
d/dx (Cos kx)/k
Sin kx
2 important trigonometric formulas
sin²x = (1 - cos2x)/2 cos²x = (1 + cos2x)/2
Sin a x cos b =
1/2 [sin(a+b) + sin(a-b)]
Cos A x sin B
1/2 x [sin(a+b) - sin(a-b)]
In a graph the area is equal to
Area of many small strips.
Area of each small strip = f(x)dx
Sum of total area of all strips = ∫f(x)dx
(Actually it is a definite integral)
d(u/v) /dx =
Quotient rule :
(v.du/dx - u.dv/dx) / v²
Cos²x - sin²x =
Cos (2x)
Derivative of tanx
Sec²x
Derivate of sec x
Sec x. tan x
Derivative of cosec x
Cosec x. Cot x
Chain rule also called
Outside inside rule.
Inverse operation of differentiation
Integration
Anti derivative of x²
(x²⁺¹) / (2+1) + C
Anti derivative of 1
x + C
3 conditions for a physical qty to be a vector
Magnitude
Direction
Should obey laws of vector algebra.
Definite integral is used for
Calculation of area of a curve.
Derivative gives us
The instantaneous rate of change of a function.
Example : if derivative of a function f(x) is 3x
It means the instantaneous rate of change of that function when x=7, would be 3x7 = 21.
Tan 53
4/3
Tan 37
3/4
