Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the derivative of $h(x)=(\frac{x^2+x}{\tan(x)})$

View step-by-step help

View step-by-step help

Find derivatives using the quotient rule I

Full solution

Q. Find the derivative of

h(x)=(\frac{x^2+x}{\tan(x)})

Identify Functions: Let's identify the two functions that are being divided. The numerator is $u(x) = x^2 + x$ and the denominator is $v(x) = \tan(x)$ . We will need to find the derivatives of both functions.
Find Derivatives: The derivative of the numerator $u(x) = x^2 + x$ is $u'(x) = 2x + 1$ .
Apply Quotient Rule: The derivative of the denominator $v(x) = \tan(x)$ is $v'(x) = \sec^2(x)$ , since the derivative of $\tan(x)$ is $\sec^2(x)$ .
Calculate $h'(x)$ : Now, we can apply the quotient rule to find the derivative of $h(x)$ . The quotient rule states that $(\frac{u}{v})' = \frac{v \cdot u' - u \cdot v'}{v^2}$ . Let's apply this to our functions.
Simplify Expression: Using the quotient rule, we get $h'(x) = \frac{\tan(x) \cdot (2x + 1) - (x^2 + x) \cdot \sec^2(x)}{\tan^2(x)}$ .
Final Derivative: We can simplify the expression by distributing and combining like terms if possible.
Final Derivative: We can simplify the expression by distributing and combining like terms if possible.However, there is no further simplification that can be done without expanding $\tan(x)$ and $\sec^2(x)$ in terms of sine and cosine, which is not necessary for finding the derivative. So, the derivative of $h(x)$ is $h'(x) = \frac{\tan(x) \cdot (2x + 1) - (x^2 + x) \cdot \sec^2(x)}{\tan^2(x)}$ .

More problems from Find derivatives using the quotient rule I

Question

Find the derivative of

f(x)

\newline

f(x) = \cos(x)

\newline

f'(x) =

Posted 1 month ago

Question

Find the derivative of

f(x)

\newline

f(x) = \tan(x)

\newline

f'(x) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = e^x

\newline

f'\left(x\right) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = \ln(x)

\newline

f'(x) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = \tan^{-1}{x}

\newline

f'(x) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = x e^x

\newline

f'\left(x\right) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = \frac{e^x}{x}

\newline

f'(x) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = e^{(x + 1)}

\newline

f'\left(x\right) =

Posted 1 month ago

Question

Find the derivative of

f(x)

\newline

f(x) = \sqrt{x+3}

\newline

f'(x) =

Posted 2 months ago

Question

Find the derivative of

g(x)

\newline

g(x) = \ln(2x)

\newline

g^{\prime}(x) =

Posted 3 months ago