Welcome to entry #5 of 21 in our Calculus Revisited series. Today, we tackle derivatives!
First the formal definition.
Derivative:
df/dx = f'(x) =
lim ( f(x + δx) - f(x) ) / δx
δx → 0
Note that both symbols for derivatives, the quotient-like symbol and the use of the prime symbol are used interchangeably.
Use the derivative to:
1. Find the instantaneous rate of change at point x0 using the function f(x).
2. Find the slope at given point.
3. When given an equation dealing with the position of an object, f(t), you can find the velocity of an object by calculating f'(t).
But we are going to jump into doing the derivatives. Here is a basic of derivatives:
d/dx ( f(x) + g(x) ) = f'(x) + g'(x)
d/dx ( f(x) * g(x) ) = f'(x) * g(x) + f(x) * g'(x)
d/dx ( f(x) / g(x) ) = (g(x) * f'(x) - g'(x) * f(x))/(g(x)^2)
d/dx a = 0 (a is a constant)
d/dx x = 1
d/dx x^n = n * x^(n-1)
d/dx sin x = cos x
d/dx cos x = -sin x
d/dx tan x = sec^2 x
d/dx e^x = e^x
d/dx a^x = a^x ln a (a is a constant)
d/dx ln x = 1/x
d/dx asin x = 1/√ (1 - x^2)
d/dx acos x = -1/√ (1 - x^2)
d/dx atan x = 1/(x^2 + 1)
TIP: For d/dx ( f(x) / g(x) ), let N = f(x) (numerator) and D = g(x) (denominator). Then d/dx N / D = ( D * N' - D' * N)/(D^2)
Higher Order Derivative: Repeat the derivative operation on f(x).
f''(x) = d^2/dx^2 is the second derivative: take the derivative of f(x), twice.
f'''(x) = d^3/dx^3 is the third derivative: take the derivative of f(x), thrice (three times).
f^(n)(x) = d^n/dx^n is the nth derivative: take the derivative of f(x) n times.
Problems
1. Find the derivative of f(x) = x^2 + 2x + 3.
We can use the addition property to help us.
d/dx (x^2 + 2x + 3)
= d/dx (x^2) + d/dx (2x) + d/dx (3)
= 2x + 2 + 0
= 2x + 2
Remember, the derivative of a constant is 0.
2. Find the slope of f(x) = -x^5 + 2x - 1 at x = 1
First find the derivative
d/dx (-x^5 + 2x - 1)
= d/dx (-x^5) + d/dx(2x) - d/dx(1)
= -5x^4 + 2 - 0
= -5x^4 + 2
The slope is determined by calculating f'(1).
f'(1) = -5(1)^4 + 2 = -3
The slope of f(x) at x = 1 is -3.
3. Find the derivative of f(x) = x^2 * ln x.
Use the product rule: x^2 multiplied by ln x.
d/dx (x^2 * ln x)
= d/dx (x^2) * ln x + x^2 * d/dx (ln x)
= 2x * ln x + x^2 * 1/x
= 2x * ln x + x
4. Find the derivative of f(x) = (x^2 -1)/(x + 2)
Use the division rule: numerator is x^2 -1 and denominator is x + 2.
d/dx ((x^2 - 1)/(x + 2))
= ((x + 2) * d/dx(x^2 - 1) - (x^2 - 1) * d/dx(x + 2))/(x + 2)^2
= ((x + 2) * 2x - (x^2 - 1) * 1)/(x + 2)^2
= (2x^2 + 4x - x^2 + 1)/(x + 2)^2
= (x^2 + 4x + 1)/(x + 2)^2
5. Higher order derivatives: Find the first, second, and third derivative of f(x) = 3x^6 + 7x and g(x) = sin x respectively.
f(x) = 3x^6 + 7x
First Derivative
f'(x) = 18x^5 + 7
Second Derivative
f''(x) = 90x^4
Third Derivative
f'''(x) = 360x^3
g(x) = sin x
g'(x) = cos x
g''(x) = -sin x
g'''(x) = -cos x
6. An interesting property: d/dx( asin x + acos x)
d/dx(asin x + acos x)
= d/dx asin x + d/dx acos x
= 1/√(1-x^2) + (-1)/√(1-x^2)
= 0
This implies that the function asin x + acos x for all x. Bonus question: what is that constant?
Thanks for joining us. Next time we will work with an important rule in calculus: The Chain Rule.
This blog is property of Edward Shore. © 2012
By the way, the answer to the bonus question: asin x + acos x = π/2