Derivative of tan^4(3x) - Solving a Simple Derivative Question

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In summary, the derivative of tan^4(3x) is 4*tan(3x)^3*sec^2(3x)(3). The teacher initially wrote 4TAN^3(x) + sec^2(3x)(3), but it appears that there was a mistake. By applying the generalized power rule, we can see that the correct formula for finding the derivative is n*u^(n-1)*u', where u(x) is the function and n is the power.
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Chocolaty
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What is the derivative of:
tan^4(3x)

The teacher wrote 4TAN^3(x) + sec^2(3x)(3) but i think he made a mistake.

I know that tan^4(3x) = tan(3x)^4 and that this equals 4tan(3x) in algebra
When I derive this i get the following: sec^2(3x)(3) I'm not sure what to do with the coefficient 4. Can someone give me the exact derivative formula to work this term out so I can copy it down please.

thanks
 
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  • #2
nevermind i got it. You need to use the generalized power rule:
[u(x)]^n => n*u^(n-1)*u'
so
tan(3x)^4 => 4*tan(3x)^3*sec^2(3x)(3)
 

FAQ: Derivative of tan^4(3x) - Solving a Simple Derivative Question

1. What is a derivative?

A derivative is a mathematical concept that represents the slope of a curve at a specific point. It measures the rate of change of a function with respect to its independent variable.

2. How do you calculate a derivative?

The standard way to calculate a derivative is by using the formula f'(x) = lim h->0 [f(x+h) - f(x)]/h, where h represents a small change in the independent variable. This formula is known as the limit definition of a derivative.

3. Why are derivatives important?

Derivatives are important because they are used to solve many real-world problems in fields such as physics, economics, and engineering. They also help us understand the behavior of a function and its rate of change at a specific point.

4. What is the difference between a derivative and an integral?

A derivative measures the slope of a function, while an integral measures the area under the curve of a function. In other words, a derivative tells us how a function is changing at a specific point, while an integral tells us the accumulated change of the function over a certain interval.

5. Can you give an example of a simple derivative question?

One example of a simple derivative question is finding the derivative of the function f(x) = 2x^2 + 3x + 1. Using the power rule and the sum rule, the derivative would be f'(x) = 4x + 3. This tells us that the slope of the original function at any point is equal to 4 times the value of x, plus 3.

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