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

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) f ( t ) = sec ( 4 t + 5 )

this problem, we're asked to find the derivative of the function f of t equals secant of four t plus five. 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 plus five. 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 gonna be here? Because you may be tempted because we're gonna 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 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 four t plus five 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 four t plus five? Well, it's just four. 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 four times the tangent of four t plus five times the secant of four t plus five. 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) f ( t ) = sec ( 4 t + 5 )

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 )

12. Trigonometric Functions

Derivatives of Trig Functions

Business Calculus

12. Trigonometric Functions

Derivatives of Trig Functions

Struggling with Business Calculus?

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Multiple Choice

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

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

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

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

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

Verified step by step guidance

Step 1: Recognize that the function f(t) = sec(4t + 5) involves the secant function, which has a derivative rule. The derivative of sec(u) with respect to t is given by d/dt[sec(u)] = sec(u)tan(u) * du/dt, where u is a function of t.

Step 2: Identify the inner function u = 4t + 5. This is the argument of the secant function. To apply the chain rule, we first need to compute the derivative of u with respect to t, which is du/dt = 4.

Step 3: Apply the chain rule. The derivative of f(t) = sec(4t + 5) is given by f'(t) = sec(4t + 5)tan(4t + 5) * (du/dt). Substitute du/dt = 4 into this expression.

Step 4: Simplify the expression. Multiply sec(4t + 5)tan(4t + 5) by 4 to get f'(t) = 4sec(4t + 5)tan(4t + 5).

Step 5: Conclude that the derivative of f(t) = sec(4t + 5) is f'(t) = 4sec(4t + 5)tan(4t + 5). This is the final simplified form of the derivative.

3:53m

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