How do I use the chain rule to find the derivative of
square root(1-x^2)
also, are there any general hints or tips for determining when the chain rule and product or quotient rule should be used?? i'm having trouble discerning when both the chain rule and either the quotient or product rules are necessary to solve the problem.
thank you.
This is often a toss-up
generally if you have a single power, such as yours, use the chain rule.
I you see a multiplication (hence product) use the product rule, etc.
at times you have to use one rule while applying another.
e.g. y = (3x+5)(2x-5)^5
is primarily a product rule, but while you are doing that you will have to use the chain rule for the power.
for your question first change the root sign to an exponent of 1/2
y = (1-x^2)^(1/2)
dy/dx = 1/2(1-x^2)^(-1/2)(2x)
= x(1-x^2)^(-1/2) or x/√(1-x^2)
To find the derivative of the function square root(1-x^2), we can use the chain rule.
The chain rule states that if we have a composite function y = f(g(x)), where f(x) is differentiable and g(x) is differentiable, then the derivative of y with respect to x, denoted as dy/dx, is given by dy/dx = f'(g(x)) * g'(x).
In our case, let's consider y = square root(1-x^2). We can rewrite this function as y = (1 - x^2)^(1/2). Now, we can identify f(u) = u^(1/2) and g(x) = 1 - x^2.
First, let's find the derivative of g(x), which is g'(x). The derivative of g(x) = 1 - x^2 with respect to x can be found by using the power rule:
g'(x) = -2x.
Next, let's find the derivative of f(u). The derivative of f(u) = u^(1/2) with respect to u can be found using the power rule:
f'(u) = (1/2) * u^(-1/2).
Now, we have f'(g(x)) and g'(x). Substituting these values into the chain rule formula, we get:
dy/dx = f'(g(x)) * g'(x)
= (1/2) * (1 - x^2)^(-1/2) * (-2x).
Simplifying this expression, we get:
dy/dx = -x / sqrt(1 - x^2).
To summarize, the derivative of square root(1-x^2) is -x / sqrt(1 - x^2).
Now, regarding your question about when to use the chain rule, product rule, and quotient rule, here are some general hints or tips:
1. Chain Rule: Use the chain rule when you have a composite function, i.e., when a function is composed of another function inside it.
2. Product Rule: Use the product rule when you have a function that can be expressed as a product of two or more functions.
3. Quotient Rule: Use the quotient rule when you have a function that can be expressed as a quotient of two functions.
It's important to carefully analyze the structure of the function and identify if it can be simplified or expressed as a composite, product, or quotient of functions. Then, apply the appropriate rule accordingly.
Practice and exposure to a variety of problems will improve your intuition for determining which rule is necessary in different situations. It can also be helpful to review the basic rules of differentiation, such as the power rule and the trigonometric rules, as they are often used in combination with the chain, product, or quotient rules.
I hope this explanation helps! Let me know if you have any further questions.