find y' of the functuion in terms of appropriate variable. simplify as far as possible.
y= 1+5^x^2-4tan(3x)
Can you confirm if the expression is meant to be:
y= [1+5^(x²)] - 4tan(3x)
y' = (ln5)(2x)(5^(x^2)) - 12 sec^2 (3x)
To find the derivative of y in terms of an appropriate variable, we will need to use the rules of differentiation.
Let's break down the function y step by step:
y = 1 + 5^(x^2) - 4tan(3x)
To find y', we will differentiate each term separately using the rules of differentiation. Let's start with each term:
1: The derivative of a constant is always 0.
So, the derivative of 1 with respect to x is 0.
5^(x^2): To differentiate this term, we will use the chain rule.
Let u = x^2
Then, y = 5^u
Using the chain rule, the derivative of y with respect to u is: dy/du = 5^u * ln(5)
Now, to differentiate with respect to x, we multiply by the derivative of u with respect to x:
dy/dx = (dy/du) * (du/dx)
dy/dx = 5^(x^2) * ln(5) * 2x
dy/dx = 2x * 5^(x^2) * ln(5)
-4tan(3x): To differentiate this term, we will use the chain rule.
Let u = 3x
Then, y = -4tan(u)
Using the chain rule, the derivative of y with respect to u is: dy/du = -4sec^2(u)
Now, to differentiate with respect to x, we multiply by the derivative of u with respect to x:
dy/dx = (dy/du) * (du/dx)
dy/dx = -4sec^2(3x) * 3
dy/dx = -12sec^2(3x)
Putting it all together, the derivative of y with respect to x (y') is:
y' = 0 + 2x * 5^(x^2) * ln(5) - 12sec^2(3x)
This is the derivative of the given function y in terms of the appropriate variable.