Find the derivative of the function.
1. F(x)= (1+2x+x^3)^1/4
2. f(t)= (1+tant)^1/3 ---> i hate trig.
3. y=4sec5x
4. y=xsin√x
To find the derivative of each function, we can use the rules of differentiation. Here's how you can find the derivatives for the given functions:
1. F(x) = (1 + 2x + x^3)^(1/4)
To find the derivative of this function, we can use the chain rule:
- Start by taking the derivative of the expression inside the parentheses (1 + 2x + x^3) with respect to x, which gives us 3x^2 + 2.
- Multiply the derivative of the expression inside the parentheses by the derivative of the exponent (1/4), which gives us (1/4) * (3x^2 + 2).
- Finally, simplify the expression if needed.
So, the derivative of F(x) is (1/4)(3x^2 + 2).
2. f(t) = (1 + tan(t))^(1/3)
To differentiate this function, we will also use the chain rule:
- Take the derivative of the expression inside the parentheses (1 + tan(t)) with respect to t, which gives us sec^2(t).
- Multiply the derivative of the expression inside the parentheses by the derivative of the exponent (1/3), which gives us (1/3) * sec^2(t).
- Since you mentioned you "hate trig," we can rewrite sec^2(t) in terms of cos(t) if you prefer. The derivative becomes (1/3) * (1/cos^2(t)).
- Simplify and rewrite the expression as needed.
Thus, the derivative of f(t) is (1/3) * (1/cos^2(t)).
3. y = 4sec(5x)
To find the derivative of this function, we can use the chain rule once again:
- Start by taking the derivative of sec(5x) with respect to x, which gives us 5sec(5x)tan(5x).
- Multiply the derivative of sec(5x) by the constant factor 4.
- Simplify the expression if needed.
Therefore, the derivative of y is 20sec(5x)tan(5x).
4. y = x * sin(√x)
To differentiate this function, we will use the product and chain rules:
- Use the product rule, which states that the derivative of the product of two functions is the first function times the derivative of the second function, plus the second function times the derivative of the first function.
- Take the derivative of x with respect to x, which is simply 1.
- Take the derivative of sin(√x) with respect to x using the chain rule. The derivative of sin(u) is cos(u), and here u = √x. So we get cos(√x) * (1/2√x).
- Multiply the first function (x) by the derivative of the second function (cos(√x) * (1/2√x)).
- Multiply the second function (sin(√x)) by the derivative of the first function (1).
- Simplify the expression if needed.
Hence, the derivative of y is sin(√x)/2√x + cos(√x).