Given values, find derivatives

  • Thread starter Jacobpm64
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In summary, to find the derivatives of g(x) and h(x) at x = 1, first we use the chain rule to find g'(x) = \frac {1}{2} f(x)^\frac{-1}{2} * f'(x) and then plug in the given values to find g'(1) = \frac {3}{4}. For h'(1), we use the fact that h(x) = f(\sqrt{x}) and plug in x = 1 to find h'(1) = 3.
  • #1
Jacobpm64
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Given y = f(x) with f(1) = 4 and f'(1) = 3, find

(a) [tex] g'(1) if g(x) = \sqrt {f(x)} [/tex]
(b) [tex] h'(1) if h(x) = f(\sqrt {x}) [/tex]

(a) [tex] g'(x) = \frac {1}{2} f(x)^\frac{-1}{2} * f'(x) [/tex]
[tex] g'(1) = \frac {1}{2} f(1)^\frac{-1}{2} * f'(1) [/tex]
[tex] g'(1) = \frac {1}{2}(4)^\frac{-1}{2} * 3 [/tex]
[tex] g'(1) = \frac {3}{4} [/tex]

(b) [tex] h'(x) = f'(\sqrt{x}) [/tex]
[tex]h'(1) = f'(\sqrt{1}) [/tex]
[tex]h'(1) = f'(1) [/tex]
[tex]h'(1) = 3 [/tex]

Are these correct?

I'm not sure if this was the correct approach.

Thanks.
 
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  • #2
(a) correct(b) If [tex] h(x) = f(\sqrt{x}) [/tex], then [tex] h'(x) = f'(\sqrt{x})\frac{1}{2}x^{-\frac{1}{2}} [/tex]. So it should be [tex] \frac{3}{2}[/tex]
 
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  • #3
thanks, just a simple mistake :redface:
 

1. What is a derivative?

A derivative is a mathematical concept that measures the rate at which a function is changing at a specific point. It represents the slope of the tangent line to the function at that point.

2. Why do we need to find derivatives?

Derivatives are used in many areas of science and mathematics, including physics, engineering, economics, and statistics. They help us understand the behavior and relationships between variables in a function, which is crucial for making predictions and solving problems.

3. How do you find derivatives?

To find the derivative of a function, we use a set of rules and formulas called the derivative rules. These rules allow us to find the derivative of different types of functions, such as polynomial, exponential, logarithmic, and trigonometric functions.

4. What are the given values in finding derivatives?

The given values refer to the specific point at which we want to find the derivative. This point is represented by a variable, usually denoted as x or t, and is often referred to as the independent variable.

5. Can we find derivatives for any function?

Yes, we can find derivatives for any continuous function. However, some functions may require more advanced techniques such as the chain rule or implicit differentiation to find the derivative.

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