# Can anyone explain the concept of derivatives?

This is a really broad question that is difficult to summarize in a few sentences.

Finding the derivatives of functions can be done in a variety of ways, with a wide range of applications. Here are some examples:

According to the power rule:

Using the product rule, we can now determine the derivative of the complete function.

Using the product rule and implicit differentiation, we obtain:

Using both sides' natural logarithms:

The line that touches the curve exactly once, at the designated point, is known as the tangent line to the curve.

Finding the line's equation in point-slope form is the final step in solving problems like these.

I'll leave it to other contributors to explain curve sketching and applications involving optimization, biological sciences, economics, and geometric figures as there are more uses for derivatives, but this answer is already very lengthy.

I hope this summary has helped you learn more about calculus and has been informative.

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Here's a visual way to approach explaining derivatives.

A derivative is the instantaneous rate of change, like at a specific point on a graph. So, it is the slope over very small changes in

Mathematically, you have:

#lim_(Deltax->0) (f(x + Deltax) - f(x))/(Deltax)#

This is really saying the same thing as

Conceptually, take a graph on a calculator, and zoom in.

- Take the idea of the slope,
#(Deltay)/(Deltax)# (rise over run), and consider*really small*#Deltax# and#Deltay# . - You can achieve
*small*values of#Deltax# and#Deltay# by zooming into a graph really closely, since your viewing window is progressively looking at smaller and smaller regions.For example,

*velocity is the derivative of position*, so zooming into a position vs. time graph gives you the instantaneous velocity within the*small*interval#[a,b]# :*When your graph looks linear at your specific spot on your graph, that represents what it means to take the derivative at that spot.*

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Derivatives are a fundamental concept in calculus that represent the rate at which a function changes with respect to one of its variables. Geometrically, the derivative of a function at a certain point corresponds to the slope of the tangent line to the curve of the function at that point. In simpler terms, derivatives measure how fast a function is changing at a specific point. They are often denoted by the symbol "d/dx" or by a prime mark (') followed by the variable with respect to which the differentiation is performed. Derivatives are used in various fields such as physics, economics, engineering, and more, to analyze rates of change, optimization problems, and to understand the behavior of functions.

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*Answer from HIX Tutor*

*When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.*

*When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.*

*When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.*

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