Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the derivative of
f(x)=8sec(x)+6x^(4)cos(x).

$9$ . Find the derivative of $f(x)=8 \sec (x)+6 x^{4} \cos (x)$ .

View step-by-step help

View step-by-step help

Find derivatives of other trigonometric functions

Full solution

Q.

9

. Find the derivative of

f(x)=8 \sec (x)+6 x^{4} \cos (x)

.

Identify Components: Identify the components of the function that need to be differentiated. $\newline$ $f(x) = 8\sec(x) + 6x^{4}\cos(x)$ consists of two terms: $8\sec(x)$ and $6x^{4}\cos(x)$ . We will need to use the sum rule for derivatives, which states that the derivative of a sum is the sum of the derivatives.
Differentiate First Term: Differentiate the first term, $8\sec(x)$ . The derivative of $\sec(x)$ is $\sec(x)\tan(x)$ , so the derivative of $8\sec(x)$ is $8\sec(x)\tan(x)$ .
Differentiate Second Term: Differentiate the second term, $6x^{4}\cos(x)$ . This term requires the product rule for derivatives, which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Let $u = 6x^{4}$ and $v = \cos(x)$ , then $\frac{du}{dx} = 24x^{3}$ and $\frac{dv}{dx} = -\sin(x)$ .
Apply Product Rule: Apply the product rule to $6x^{4}\cos(x)$ . Using the product rule, the derivative of $6x^{4}\cos(x)$ is $(\frac{du}{dx})v + u(\frac{dv}{dx}) = (24x^{3})\cos(x) + 6x^{4}(-\sin(x))$ .
Combine Derivatives: Combine the derivatives of both terms. $\newline$ The derivative of $f(x)$ is the sum of the derivatives from Step $2$ and Step $4$ . Therefore, $f'(x) = 8\sec(x)\tan(x) + (24x^{3})\cos(x) - 6x^{4}\sin(x)$ .

More problems from Find derivatives of other trigonometric functions

Question

Find the derivative of

f(x)

\newline

f(x) = \cos(x)

\newline

f'(x) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = \tan(x)

\newline

f'(x) =

Posted 3 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = e^x

\newline

f'\left(x\right) =

Posted 3 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = \ln(x)

\newline

f'(x) =

Posted 3 months ago

Question

Find the derivative of

f(x)

\newline

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

\newline

f'(x) =

Posted 3 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = x e^x

\newline

f'\left(x\right) =

Posted 3 months ago

Question

Find the derivative of

f(x)

\newline

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

\newline

f'(x) =

Posted 3 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 3 months ago

Question

Find the derivative of

g(x)

\newline

g(x) = \ln(2x)

\newline

g^{\prime}(x) =

Posted 4 months ago