1) y=4+5x + 7x³. Find dy/dx
2) if y= tan x cos²x then dy by dx will be-
1) 0 + 5 + 21 x^2
2) tan x (d cos^2 x /dx)+ cos^2 x d tan x / dx
= tan x (-2 sin x cos x) + cos^2 x (sec^2 x)
= -2 sin^2 x + 1
To find the derivative of a function, such as dy/dx, you need to use calculus rules. In this case, we'll use the power rule and the chain rule.
1) To find dy/dx for y = 4 + 5x + 7x³, we first need to differentiate each term separately.
- For the constant term 4, its derivative is 0 since the derivative of a constant is always 0.
- For the term 5x, its derivative is simply 5 since the derivative of x with respect to x is 1.
- For the term 7x³, we'll use the power rule. The power rule states that if you have a term of the form ax^n, the derivative is nax^(n-1). So the derivative of 7x³ will be 3 * 7 * x^(3-1) = 21x².
Summing up the derivatives of each term, dy/dx = 5 + 21x².
2) To find dy/dx for y = tan(x) * cos²(x), we'll need to use the product rule and chain rule.
The product rule states that if you have two functions, f(x) and g(x), the derivative of their product is given by: (f'(x) * g(x)) + (f(x) * g'(x)).
The chain rule states that if you have a composition of functions, such as f(g(x)), the derivative is given by: f'(g(x)) * g'(x).
Let's break down the equation y = tan(x) * cos²(x) into its constituent parts:
f(x) = tan(x)
g(x) = cos²(x)
Now, we'll find the derivatives of f(x) and g(x):
f'(x) = sec²(x) (derivative of tan(x) is sec²(x))
g'(x) = 2cos(x)(-sin(x)) (derivative of cos²(x) using the chain rule)
Applying the product rule, dy/dx = (f'(x) * g(x)) + (f(x) * g'(x)):
dy/dx = (sec²(x) * cos²(x)) + (tan(x) * 2cos(x)(-sin(x)))
Simplifying the expression further if needed.
Please note that these derivatives are calculated based on the given equations.