Find the derivative of the function.f(t)=sec(4t+5)f\left(t\right)=\sec\left(4t+5\right)

4 sec ⁡ ( 4 t + 5 ) tan ⁡ ( 4 t + 5 ) 4\sec\left(4t+5\right)\tan\left(4t+5\right) 4 sec ( 4 t + 5 ) tan ( 4 t + 5 )

Find the derivative of the function. f ( t ) = sec ( 4 t + 5 ) f\left(t\right)=\sec\left(4t+5\right)

this problem, we're asked to find the derivative of the function f of t equals secant of 4 t+5. Now since I have a function inside of the argument of that trig function, that tells me I need to use the chain rule here because I have a function inside of another function. So I have this inside function for t+5. My outside function is that secant. So let's go ahead and find our derivative here, f prime of t. Now starting with that outside function, working our way inside, our outside function is the secant. What is the derivative of secant? Well, it's tangent times secant. Now, what exactly is our argument going to be here? Because you may be tempted because we're going to have to write multiple arguments because I have the tangent and the secant to kind of mess this up and get a little bit confused. It can be really easy to get mixed up in problems like this. But remember that when working with our chain rule, if we're working with our outside function, our inside function stays the same. We're effectively treating it like a variable, meaning that when I take this derivative and I get the tangent times the secant, I want my argument to remain the same. It's that 4t+5 that we have up there as our inside function. We don't wanna touch it at this point. It's staying the same. So that is our first part of our chain rule done. We've done our outside function. Now we're gonna multiply this by the derivative of that inside function. We are still working our way inside. So what is the derivative of 4t+5? Well, it's just 4. So my final answer here, if I move that constant up to the front just to match convention, make this look a little bit nicer, I have 4 times the tangent of 4t+5 times the secant of 4t+5. So this is my final answer here for my derivative and we are all done here. Let me know if you have any questions and let's continue learning.

Find the derivative of the function. f ( t ) = sec ⁡ ( 4 t + 5 ) f\left(t\right)=\sec\left(4t+5\right)

4 sec ⁡ ( 4 t + 5 ) tan ⁡ ( 4 t + 5 ) 4\sec\left(4t+5\right)\tan\left(4t+5\right) 4 sec ( 4 t + 5 ) tan ( 4 t + 5 )

4 sec ⁡ ( 4 t + 5 ) 4\sec\left(4t+5\right) 4 sec ( 4 t + 5 )

4 sec ⁡ ( t ) tan ⁡ ( t ) 4\sec\left(t\right)\tan\left(t\right) 4 sec ( t ) tan ( t )

sec ⁡ ( 4 ) tan ⁡ ( 4 ) \sec\left(4\right)\tan\left(4\right) sec ( 4 ) tan ( 4 )

3. Techniques of Differentiation

The Chain Rule

Calculus

3. Techniques of Differentiation

The Chain Rule

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Multiple Choice

Find the derivative of the function.
$f (t) = \sec (4 t + 5)$

$4\sec\left(4t+5\right)$

$4\sec\left(4t+5\right)\tan\left(4t+5\right)$

$4\sec\left(t\right)\tan\left(t\right)$

$\sec\left(4\right)\tan\left(4\right)$

Verified step by step guidance

Identify the function for which you need to find the derivative: $ f(t) = \sec(4t + 5) $.

Recall the derivative rule for the secant function: if $ f(x) = \sec(u) $, then $ f'(x) = \sec(u) \tan(u) \cdot u' $, where $ u $ is a function of $ x $.

In this problem, $ u = 4t + 5 $. First, find the derivative of $ u $ with respect to $ t $, which is $ u' = \frac{d}{dt}(4t + 5) = 4 $.

Apply the chain rule: the derivative of $ f(t) = \sec(4t + 5) $ is $ f'(t) = \sec(4t + 5) \tan(4t + 5) \cdot 4 $.

Simplify the expression to get the final derivative: $ f'(t) = 4 \sec(4t + 5) \tan(4t + 5) $.

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