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Find the equation of the tangent line to the curve at the given $x$ -value.$$\text { Compute: } \quad(a) \frac{d}{d y}\left(y^{7}\right), \quad(b) \frac{d}{d x}\left(y^{7}\right), \quad(c) \frac{d}{d x}\left(\left(x^{5}+1\right) y^{7}\right)$$

(a) $7 y^{6}$(b) $7 y^{6} \frac{d y}{d x}$(c) $7\left(x^{5}+1\right) y^{6} \frac{d y}{d x}+5 x^{4} y^{7}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 6

The Chain Rule

Derivatives

Oregon State University

Baylor University

University of Michigan - Ann Arbor

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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eso as I'm reading through these problems, I'm just thinking of implicit differentiation. So pay attention to the variable. So D d y of why do the seventh power? Okay, we're good to do this with our old rules. Where you bring why in front and then Sorry, the seven in front. Then why To the six power could you subtract the exponents? But then as I moved apart, B As you see, it says d d x. So the independent variable is x of why to the seventh? Well, there's no X to take the derivative of So what you do is you. It's kind of like implicit differentiation. Uh, where you bring that seven in front? Why? To the six. Sorry, this is a chain rule. Or now you have to take the derivative of y. Well, what is the drift of y? It's the y dx, um and it's all based off of the independent variable. So this is the answer we're looking for on B. And then as you look at sea, um, we're doing the derivative of this quantity X to the fifth plus one times why in the seventh power. So notice that we have the product will run on. And the general well, the product, we would say Take the derivative of the first product, which would be just five X to the fourth, the derivative of 10 So nothing right there and you leave the second half. Why did the seventh alone? But then plus now you leave X to the fifth plus one alone times the derivative of y to the seventh. Well, that's seven y to the six. And then because we're taking the derivative with respect to X to take the derivative of y. So it's implicit differentiation. If you've seen that before, otherwise, you just pay attention to the variable that we're differentiating. And these were three answers that one A B and this is seen. If you want to, you can simplify this, or at least you could distribute in here because you don't see the benefit of doing so. That's why I'm leaving my answer like this

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