- #1
helpm3pl3ase
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(sec4x + 4arctanx^2)=
(sec4x)(tan4x) + (4)(1/1+x^4)(2x).. Did I derive this correctly??
(sec4x)(tan4x) + (4)(1/1+x^4)(2x).. Did I derive this correctly??
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.
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.
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.
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.
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.