Chain rule with leibniz notation

In summary, using the chain rule, we can find dy/dx when x=4 for a function y=f((x2+9)0.5) given that f'(5)=-2. However, the given information of f'(5) is not necessary to solve the problem. Using the chain rule, we can find that dy/dx = x/((x2+9))0.5. Plugging in the value of x=4, we get dy/dx(4) = 4/(25)0.5 = 4/5, which gives us the final answer of -8/5.
  • #1
[ScPpL]Shree
3
0

Homework Statement



If y=f((x2+9)0.5) and f'(5)=-2, find dy/dx when x=4

Homework Equations



chain rule: dy/dx=(dy/du)(du/dx)

The Attempt at a Solution



In my opinion giving f'(5)=-2 is unnecessary as:

y=f(u)=u, u=(x2+9)0.5

dy/dx= (dy/du)(du/dx)

(dy/du)= 1
(du/dx)= x/((x2+9))0.5

dy/dx = (1)(x/((x2+9))0.5)
= x/((x2+9))0.5
dy/dx(4) = 4/(16+9)0.5
= 4/(25)0.5
= 4/5

the answer is -8/5

I would appreciate help very much.
 
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  • #2
wait, how do you get f(u)=u?
 
  • #3
[ScPpL]Shree;2830588 said:

Homework Statement



If y=f((x2+9)0.5) and f'(5)=-2, find dy/dx when x=4

Homework Equations



chain rule: dy/dx=(dy/du)(du/dx)

The Attempt at a Solution



In my opinion giving f'(5)=-2 is unnecessary as:

y=f(u)=u, u=(x2+9)0.5
As Annoymage pointed out, you can't assume that f(u) = u.
[ScPpL]Shree;2830588 said:
dy/dx= (dy/du)(du/dx)

(dy/du)= 1
y = f(u), so dy/dy = f'(u)
[ScPpL]Shree;2830588 said:
(du/dx)= x/((x2+9))0.5

dy/dx = (1)(x/((x2+9))0.5)
= x/((x2+9))0.5
dy/dx(4) = 4/(16+9)0.5
= 4/(25)0.5
= 4/5

the answer is -8/5

I would appreciate help very much.
 

What is the Chain Rule with Leibniz Notation?

The chain rule with Leibniz notation is a method for finding the derivative of a composite function. It is used when a function is composed of two or more functions.

How do you use the Chain Rule with Leibniz Notation?

To use the chain rule with Leibniz notation, you first identify the outer function and the inner function. Then, you take the derivative of the outer function and multiply it by the derivative of the inner function. Finally, you substitute the inner function with its original expression.

What is the formula for the Chain Rule with Leibniz Notation?

The formula for the chain rule with Leibniz notation is d(uv)/dx = (du/dx)(dv/dx).

Why is the Chain Rule with Leibniz Notation important?

The chain rule with Leibniz notation is important because it allows us to find the derivative of complex functions that are composed of simpler functions. It is a fundamental tool in calculus and is used in many applications, such as physics, engineering, and economics.

Can you provide an example of using the Chain Rule with Leibniz Notation?

Yes, for example, if we have the function f(x) = (x^2 + 1)^3, we can rewrite it as f(x) = u^3, where u = x^2 + 1. Using the chain rule with Leibniz notation, we can find the derivative as d/dx[(x^2 + 1)^3] = 3u^2 * d/dx[(x^2 + 1)] = 3(x^2 + 1)^2 * 2x = 6x(x^2 + 1)^2.

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