Today we cover a very useful rule in Calculus: The Chain Rule. The rule deals with so many applications in higher mathematics. This is entry #6 in a series of 21.
The Chain Rule
d/dx f(g(x)) = f'(g(x)) * g'(x) = df/dg * dg/dx
If you learn nothing else in calculus - learn this rule! We will do some practice problems with chain rule today.
1. d/dx (3x^2 + 1)^2
Using the rule above, let f = g^2 and g = 3x^2 + 1.
Then d/dx f(g) = df/dg * dg/dx,
f = g^2
df/dg = 2g
g = 3x^2 + 1
dg/dx = 6x
Putting it all together:
d/dx (3x^2 + 1)^2 = 2g * 6x = 2(3x^2 + 1) * 6x = 36x^3 + 12x
2. d/dx cos(2x^3 - 1)
Let f = cos g and g = 2x^3 -1
df/dg = -sin g
dg/dx = 6x^2
Put it together:
d/dx cos(2x^3 - 1) = -sin g * 6x^2 = -sin(2x^3 -1) * 6x^2 = -6x^2 * sin(2x^3 - 1)
3. d/dx √(e^x + 1)
Note √x = x^(1/2)
Let f = √g and g = e^x + 1
df/dg = 1/2 * 1/√g
dg/dx = e^x
d/dx √(e^x + 1) = 1/2 * 1/√g * e^x = e^x / (2 * √(e^x + 1))
Now that you get the though process, let's try another one.
4. d/dx cos^3 x = d/dx (cos x)^3
d/dx (cos x)^3 = 3 * (cos x)^2 * d/dx cos x = 3 * (cos x)^2 * -sin x = - 3 cos^2 x sin x
5. Evaluate the slope of the function y(x) = ln sin x at x = π/4
d/dx ln sin x
= 1 / sin x * d/dx (sin x)
= 1 / sin x * cos x
= cot x
To find the slope at x = π/4, find cot π/4 = 1.
6. The Chain Rule can be applied more than once. This problem illustrates it. Find d/dx ln (cos √x)
d/dx ln (cos √x)
= 1/(cos √x) * d/dx (cos √x)
= 1/(cos √x) * (-sin √x) * d/dx (√x)
= (-sin √x)/(cos √x) * 1/(2√x)
= (-tan √x)/(2√x)
Next time, we will find the extremes of functions.
This blog is property of Edward Shore. © 2012