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 a function, you can usually apply the rules of differentiation. Here's how you can find the derivatives of these functions:
1. F(x)= (1+2x+x^3)^1/4:
To differentiate this function, you can use the chain rule. Recall that the chain rule states that if you have a composition of functions, such as f(g(x)), the derivative of f(g(x)) is f'(g(x)) * g'(x).
Let's apply the chain rule here:
- First, differentiate the inner function (1+2x+x^3) with respect to x, which gives you 2 + 3x^2.
- Then, apply the power rule by multiplying the derivative by the exponent: (1/4) * (2 + 3x^2) = (1/2 + 3x^2/4) = 1/2 + (3/4)x^2.
So the derivative of F(x) = (1+2x+x^3)^1/4 is 1/2 + (3/4)x^2.
2. f(t)= (1+tan(t))^1/3:
To differentiate this function, we can again use the chain rule.
- First, differentiate the inner function (1+tan(t)) with respect to t. The derivative of tan(t) is sec^2(t).
- Then, apply the power rule by multiplying the derivative by the exponent: (1/3) * sec^2(t) = sec^2(t)/3.
However, you mentioned that you dislike trigonometric functions. So, if you want to avoid trigonometric functions in your derivative, you can simplify the expression further before differentiating using the identity 1 + tan^2(t) = sec^2(t):
f(t) = (1 + tan(t))^1/3
= ((1 + tan(t))/(sec^2(t)))^(1/3)
= (sec^2(t)/(sec^2(t)))^(1/3)
= (1/1)^(1/3)
= 1
Therefore, the derivative of f(t) = (1+tan(t))^1/3 is 0.
3. y=4sec(5x):
To differentiate this function, you can use the chain rule.
- First, differentiate the inner function sec(5x) with respect to x. The derivative of sec(u) is sec(u)tan(u), where u = 5x.
- Then, apply the chain rule: d/dx sec(5x) = sec(5x)tan(5x) * (d/dx 5x) = sec(5x)tan(5x) * 5.
- Multiply the above expression by the constant 4: d/dx (4sec(5x)) = 4 * sec(5x)tan(5x) * 5.
So the derivative of y = 4sec(5x) is 20sec(5x)tan(5x).
4. y = xsin(√x):
To differentiate this function, you can use the product rule.
- Apply the product rule by differentiating x with respect to x, which gives 1, and leaving sin(√x) as it is.
- Differentiate sin(√x) with respect to x using the chain rule. The derivative of sin(u) is cos(u), where u = √x. Then multiply by the derivative of the inner function: d/dx sin(√x) = cos(√x) * (d/dx √x) = cos(√x) * (1/(2√x)).
Now, we can write the derivative of y = xsin(√x) as:
dy/dx = (1)(sin(√x)) + (x)(cos(√x))(1/(2√x))
= sin(√x) + (xcos(√x))/(2√x).
Therefore, the derivative of y = xsin(√x) is sin(√x) + (xcos(√x))/(2√x).