Derivative: sec4x tan4x + 8x/(1+x^4)

• helpm3pl3ase
In summary, a derivative is a mathematical concept that measures the instantaneous rate of change of a function. It can be found by using the rules of differentiation, such as the limit of the difference quotient or the chain rule, product rule, or quotient rule for more complex functions. The derivative is significant in mathematics for applications such as optimization and modeling real-world systems. It can also be negative, indicating a decreasing function or a concave down shape for the function's graph.
helpm3pl3ase
(sec4x + 4arctanx^2)=

(sec4x)(tan4x) + (4)(1/1+x^4)(2x).. Did I derive this correctly??

Did you use the chain rule on the sec(4x) term?

you forgot to multiply by $$du = 4$$ in the first term

(sec4x)(4)(tan4x)(4) + (4)(1/1+x^4)(2x)

or

16(sec4x)(tan4x) + (4)(1/1+x^4)(2x)??

or do i keep just 1 4?? like this..

(sec4x)(tan4x)(4) + (4)(1/1+x^4)(2x)

It would only be one four. Let sec(4x)=sec(u).

Then d(sec4x)/dx = d(secu)/dx = secu*tanu*du/dx

du/dx = 4, so

sec(4x)' = 4sec(4x)tan(4x)

1. What is a derivative?

A derivative is a mathematical concept that represents the instantaneous rate of change of a function with respect to its independent variable. In other words, it measures how much a function's output changes for a given change in its input.

2. How do you find the derivative of a given function?

The derivative of a function can be found by using the rules of differentiation, which involve taking the limit of the difference quotient as the change in the independent variable approaches zero. For more complicated functions, the chain rule, product rule, or quotient rule may need to be applied.

3. What is the derivative of sec4x tan4x + 8x/(1+x^4)?

The derivative of this function is 4(sec4x)^2 tan4x + (8 - 32x^4)/(1+x^4)^2. This can be found by applying the chain rule and product rule.

4. What is the significance of the derivative in mathematics?

The derivative has many applications in mathematics, including optimization, finding critical points and inflection points, and graphing functions. It is also used in physics and engineering to model rates of change in real-world systems.

5. Can the derivative be negative?

Yes, the derivative can be negative. This means that the function is decreasing at that point, or that the slope of the tangent line is negative. A negative derivative also indicates a concave down shape for the function's graph.

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