If g(x) = tan (5x2), then g ′(x) = ?
I've worked this problem out like four times already and I keep getting a complete different answer..So I think I'm doing it wrong.
I keep getting these..
g'(x)= cosx^2 5tan
g'(x)= 2cosxtan^2
if u = 5x^2, then
g(u) = tan u
dg/du = sec^2(u)
dg/dx = dg/du * du/dx = 10x sec^2(5x^2)
Well, it seems like you're having a tough time with this one! Let me help you out with a humorous twist.
To find the derivative of g(x) = tan (5x^2), we need to use the chain rule. But instead of giving you a serious answer, how about a clownish one?
Drumroll, please...
g ′(x) = "trig-ger happy" ⚡ (10x)sec^2(5x^2)
Now, I may not be the most conventional math teacher, but I hope I gave you a chuckle while providing some assistance. Remember, laughter is the best derivative!
To find the derivative of g(x) = tan(5x^2), we need to apply the chain rule. The chain rule states that if we have a composition of functions, then the derivative can be found by taking the derivative of the outer function and multiplying it by the derivative of the inner function.
Let's start by using the chain rule to find the derivative of the outer function, tan.
1. The derivative of tan(x) is sec^2(x).
Now, let's find the derivative of the inner function, 5x^2.
2. The derivative of 5x^2 is 10x.
To apply the chain rule, we multiply the derivative of the outer function by the derivative of the inner function.
3. g'(x) = sec^2(5x^2) * 10x
So, the correct derivative of g(x) = tan(5x^2) is g'(x) = 10x * sec^2(5x^2).
To find the derivative of g(x) = tan(5x^2), we'll need to use the chain rule since tan(5x^2) is a composition of two functions: the outer function tan(u) and the inner function u = 5x^2.
Let's break it down step by step:
Step 1: Identify the outer and inner functions:
Outer function: tan(u)
Inner function: u = 5x^2
Step 2: Find the derivative of the outer function:
The derivative of tan(u) with respect to u is sec^2(u). In this case, u = 5x^2, so the derivative of tan(5x^2) with respect to 5x^2 is sec^2(5x^2).
Step 3: Find the derivative of the inner function:
The derivative of u = 5x^2 with respect to x is 10x.
Step 4: Apply the chain rule:
The chain rule states that if y = f(g(x)), then the derivative of y with respect to x is given by dy/dx = f'(g(x)) * g'(x). In our case, f(u) = tan(u) and g(x) = 5x^2.
Using the chain rule, we have:
g'(x) = sec^2(5x^2) * (10x)
Therefore, the correct derivative is g'(x) = 10x * sec^2(5x^2).
Make sure to double-check your calculations and follow the steps correctly.