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Use the appropriate rules to determine the derivative.$$f(x)=\sqrt[3]{x}-\sqrt[4]{x}+7 x+32 x^{4}+71, \text { find } f^{\prime}(x)$$

$$\frac{1}{3 \sqrt[3]{x^{2}}}-\frac{1}{4 \sqrt[4]{x^{3}}}+7+128 x^{3}$$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 2

Derivatives Rules 1

Derivatives

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Lectures

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, eso We're going to find the derivative by just looking at rational exponents. So it makes the most sense to rewrite all of your problems as rational expose. So the cube root is to the one third power. The fourth route is to the 1/4 power. The seven X is fine the way it is. Seven next to the first might be helpful. 32 x to the fourth and plus 71. All right, so if you need help with rational exponents, there you have it. So I would on Lee take the derivative and just leave my answer as is. For instance, bring one third in front and it becomes X to the negative two thirds power when you subtract one from your expert. If you're struggling with subtracting one from exponents, just get practiced with the same denominator. So one minus three is negative Two. You know, if I do the next 1 1/4 minus one to 1/4 minus four force. So when you see that you know, comes a little bit more naturally. Yes, the dread of the seven x seven. If you need help with 32 times four Well, three times four is 12, um, 2, 10, 48 128 x to the third and then the derivative of a constant is zero eso No need to write that plus zero doesn't change anything. I would leave my answer like this. I know some other people. Other teachers might ask you to rewrite the problem as you know, 1/3 cube root of X squared. But this is all personal preference at this point. Fourth root of X cubed, you know, nothing to do with these other two things. That's good as is.

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