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Use the appropriate rules to determine the derivative.Compute $$\frac{d}{d x}\left(3 x^{5}-2 x^{3 / 2}-\frac{7}{x^{5}}+\sqrt{93}\right)$$

$$15 x^{4}-3 x^{1 / 2}+\frac{5}{x^{6}}$$

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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Yeah, So don't get confused by the notation. It's just ah, notation and say, Hey, take the derivative of the following So it saves a person from having to rewrite the equation over and over again. If 97 x to the fifth and plus Route 93 Now, before moving on, I wanna make two distinguishing things. First of all, that this is the same thing is saying minus seven x to the negative fifth power. It's gonna make my math easier. Um, and then this. I'm gonna make the distinction that that's just a constant, you know, even though you could rewrite that 93 to the one half hour, it's always the same number. It's constant and the derivative of a constant zero, which will take me to my next point. So don't forget, you may move the number in front, you know, three times five and you subtract one from that exponents. Um, Now most people should be able to do that in your head. You know, three times five is 15 x to the fourth. Same thing with the next one. You're multiplying by that three halves. These twos will canceled and you subtract one from that exponents. Now, if you're not that good at exponents yet you might wanna change one to be two halves. You know, that might make the problem easier for you. Um, again, multiplying with these rational exponents makes the problem easier. So it becomes plus 35 7 times five x to the negative six. Because you subtract one from that exponents, Um, and then the derivative of a constant is zero. So some teachers will let you leave your answer. You know, pretty pretty simple like this. You know, other teachers might say, Go ahead and rewrite the one half power as the square root of X or, you know, 35 over a negative exponents putting the denominator, you know? So this is how I would write my answer. But there are other acceptable answers as well.

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