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I’m in AP Calculus AB. Can you help me solve this showing steps?I’m not exactly sure how to do this.

I’m in AP Calculus AB. Can you help me solve this showing steps?I’m not exactly sure-example-1
User Tadasz
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1 Answer

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We need to find the derivative of a function, but if we look at the function is a quotient. Then we can use the rule for the derivative of a quotient:


\begin{gathered} f(x)=(g(x))/(h(x)) \\ \text{Then,} \\ (df)/(dx)=(g^(\prime)h-h^(\prime)g)/(h^2) \end{gathered}

In this case, we have:


f(x)=(\csc (x))/(x^2)

We can call:


\begin{gathered} g(x)=\csc (x) \\ h(x)=x^2 \end{gathered}

Now we apply the rule:


(df)/(dx)=((dg)/(dx)h-(dh)/(dx)g)/(h^2)

We know that the derivative of h(x) = x^2 is 2x, and the derivative of g(x) = csc(x) is [-cot(x)csc(x)]

Replacing the values:


(df)/(dx)=(-\cot (x)\csc (x)\cdot x^2-2x\cdot\csc (x))/((x^2)^2)

Simplifying:

We can factor out csc(x)*x in the numerator and solve the square in the denominator


(df)/(dx)=(x\csc (x)(-x\cot (x)-2))/(x^4)

Finally, we can factor the (-1) inside the parentheses and cancel out the x in the numerator:


(df)/(dx)=-(\csc (x)(x\cot (x)+2))/(x^3)

And this is the final answer.

User Chris Talman
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