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Use the appropriate rules to determine the derivative.$$f(x)=2 x^{7}-\frac{3}{x^{4}}+\frac{7}{x^{3 / 4}}+5, \text { find } f^{\prime}(x)$$

$$14 x^{6}+\frac{12}{x^{5}}-\frac{21}{4 x^{7 / 4}}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 2

Derivatives Rules 1

Derivatives

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04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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Find the derivative of the…

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Find the second derivative…

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find the second derivative…

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Find $f^{\prime \prime \pr…

04:13

find the derivative ?

So the best way to approach this problem is to write everything with, I guess, the rational exponents. So we're looking at two x of the seventh. Instead of writing three over X to the fourth toe, go ahead and write as three x to the negative fourth, uh, plus seven x to the negative three Force power, and then that plus five is sort of pointless because it's a constant and the derivative of a constant is zero so we can jump right into the derivative where you move the Expo Nen front. Multiply. So two times seven is 14 and you subtract one from your exponents. Same thing in the next one. A negative times. The name is positive. Three times four is 12. When you subtract one from negative four, you get negative five. Now the next one is kind of an ugly number because seven times three is 21 force. Uh, that's your denominator. Now, students typically struggle with subtracting one from a fraction, but just make it 44 set equals one. So we're looking at negative seven force and already said the derivative of a constant zero. Um, just most teachers would ask you to clean this up. Some teachers will let you leave your answer like this, but others would say, you know, go ahead and rewrite it back in the correct standard form. You know, 12 over X to the fifth and 21 over for X to the Seventh Force. Okay, but yeah, that's up to your teacher.

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