# How to calculate Derivative of sin sq. root x by definition?

• kashan123999
In summary, the conversation is about finding the derivative of sin sq. root x with respect to x. The attempted solution involves using the limit definition of the derivative, but the person is unsure how to proceed. They then try to rewrite the equation using trigonometric identities, but are confused about where the term (1+ Dx/sq.root x) comes from. The conversation ends with a suggestion to multiply the argument within the sine and simplify.
kashan123999

## Homework Statement

Evaluate derivative of (sin sq. root x) w.r.t x?

## Homework Equations

Limit Δx--> 0 (sin√(x+Δx) - sin(√x)) / Δx

## The Attempt at a Solution

i couldn't operate it from here... Δy = (2cos((√x+Δx) + (√x)) . sin((√x+Δx) - (√x)) / Δx...?

Rewrite:
$$\sin(\sqrt{x+Dx})=\sin(\frac{\sqrt{x}}{\sqrt{1+\frac{Dx}{\sqrt{x}}}} +\frac{\frac{Dx}{\sqrt{x}}}{\sqrt{1+\frac{Dx}{\sqrt{x}}}})$$

Last edited:
where did that sq. root (1+ Dx/sq.root x) come from?

Multiply the argument within the sine as follows:
$$\sqrt{x+Dx}=\sqrt{x+Dx}*1=(\sqrt{x+Dx})*\frac{\sqrt{x+Dx}}{\sqrt{x+Dx}}=\frac{x+Dx}{\sqrt{x+Dx}}$$
Now, extraxt "x" from the square root in the denominator and simplify.

## 1. What is the definition of a derivative?

The derivative of a function f(x) at a point x is defined as the limit of the ratio of the change in the function value to the change in the input variable, as the change in the input variable approaches zero.

## 2. What is the formula for calculating the derivative of sin sq. root x?

The formula for calculating the derivative of sin sq. root x is
f'(x) = (1/2)*cos(sqrt(x))/sqrt(x).

## 3. How do you use the definition of a derivative to calculate the derivative of sin sq. root x?

To calculate the derivative of sin sq. root x using the definition, you need to find the limit of the difference quotient as the change in x approaches 0. This involves plugging in h=0 and solving for the derivative using algebraic manipulations.

## 4. What are the steps for calculating the derivative of sin sq. root x?

The steps for calculating the derivative of sin sq. root x are:
1. Write out the definition of a derivative.
2. Substitute sin sq. root x into the definition.
3. Simplify and manipulate the expression to get it in a form that can be evaluated at h=0.
4. Take the limit as h approaches 0 to find the derivative.

## 5. Can the derivative of sin sq. root x be simplified further?

Yes, the derivative of sin sq. root x can be simplified further to f'(x) = cos(sqrt(x))/(2*sqrt(x)). This can be done by using trigonometric identities and algebraic manipulations.

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