Differentiate x-3+3tanx
1+3/cos(x)^2
boy, you missed it twice!
y = x-3+3tan(x)
y' = 1 + 3sec^2(x)
no powers or chain rule involved.
You are correct, and I apologize for the repeated errors.
The derivative of x with respect to x is 1.
The derivative of -3 with respect to x is 0 (since it is a constant).
The derivative of tan(x) with respect to x is sec^2(x).
Therefore, the derivative of x-3+3tan(x) with respect to x is:
1 + 3sec^2(x).
I apologize for the confusion caused by my previous incorrect responses. Thank you for pointing it out.
To differentiate the expression x - 3 + 3tan(x), we will use the basic rules of differentiation.
Step 1: Differentiate x with respect to x
The derivative of x with respect to x is 1.
Step 2: Differentiate -3 with respect to x
The derivative of a constant is always 0.
Step 3: Differentiate 3tan(x) with respect to x
To differentiate tan(x), we will use the chain rule.
Recall that the chain rule states:
d(u/v)/dx = (v * du/dx - u * dv/dx) / (v^2)
Let u = 3 and v = tan(x).
Then, du/dx = 0 (since 3 is a constant) and dv/dx = sec^2(x) (the derivative of tan(x)).
Applying the chain rule:
d(3tan(x))/dx = (tan(x) * 0 - 3 * sec^2(x)) / (tan^2(x))
Simplifying further, we get:
d(3tan(x))/dx = -3sec^2(x) / tan^2(x)
Step 4: Combine the derivatives obtained in steps 1, 2, and 3
Putting it together, the derivative of x - 3 + 3tan(x) is:
1 + 0 - 3sec^2(x) / tan^2(x)
Thus, the derivative of x - 3 + 3tan(x) is 1 - 3sec^2(x) / tan^2(x).
To differentiate the expression x - 3 + 3tan(x), we can use the rules of differentiation.
The expression consists of three terms: x, -3, and 3tan(x).
The first term, x, is a polynomial term that has a power of 1. The derivative of x^n, where n is a constant, is simply n*x^(n-1). In this case, the derivative of x is 1.
The second term, -3, is a constant term. The derivative of a constant is zero, as it does not depend on the variable x.
The third term, 3tan(x), involves a trigonometric function. To differentiate this term, we can use the chain rule. The chain rule states that for a composition of functions, the derivative of the outer function multiplied by the derivative of the inner function is equal to the derivative of the composition.
The derivative of tan(x) with respect to x is sec^2(x).
So, applying the chain rule, we multiply the derivative of the outer function, which is 3, by the derivative of the inner function, which is sec^2(x), and we get 3 * sec^2(x).
Putting all the terms together, the derivative of x - 3 + 3tan(x) is 1 + 0 + 3 * sec^2(x), which simplifies to 1 + 3sec^2(x).
To differentiate the expression x-3+3tan(x), we will use the sum/difference rule and the chain rule.
The derivative of x with respect to x is 1.
The derivative of -3 with respect to x is 0 (since it is a constant).
The derivative of 3tan(x) with respect to x requires the use of the chain rule.
Recall that the derivative of tan(x) with respect to x is sec^2(x).
Using the chain rule, the derivative of 3tan(x) with respect to x is:
3 * sec^2(x) * (d/dx)(tan(x)).
Applying the chain rule again, we have:
3 * sec^2(x) * (d/dx)(tan(x)) = 3 * sec^2(x) * sec^2(x).
Simplifying further, we have:
3 * sec^4(x).
Therefore, the derivative of x-3+3tan(x) with respect to x is:
1 + 0 + 3 * sec^4(x).
So, the final answer is:
1 + 3 * sec^4(x).
My apologies for the previous response. Let's differentiate the expression x-3+3tan(x) correctly.
The derivative of x with respect to x is 1.
The derivative of -3 with respect to x is 0 (since it is a constant).
The derivative of 3tan(x) with respect to x requires the use of the chain rule.
Recall that the derivative of tan(x) with respect to x is sec^2(x).
Using the chain rule, the derivative of 3tan(x) with respect to x is:
3 * sec^2(x) * (d/dx)(tan(x)).
Applying the chain rule again, we have:
3 * sec^2(x) * (d/dx)(tan(x)) = 3 * sec^2(x) * sec^2(x).
Simplifying, we have:
3 * sec^4(x).
Therefore, the derivative of x-3+3tan(x) with respect to x is:
1 + 0 + 3 * sec^4(x).
So, the final answer is:
1 + 3 * sec^4(x).