I am having trouble solving these. Could you explain how to do these 3 questions, step by step? That would be helpful!
1. Differentiate y=sin ^-1 (x^2) at x=0.5 . Round to 3 decimal points.
3. Differentiate y= (x^2 +1) tan ^-1 (x) at x=1
7. Differentiate y= [cos^-1 (x)]^3 at x=0.5
1. y = arcsin(x ²)
Let u = x ²
y = arcsin(u)
dy/dx = (dy/du)(du/dx)
dy/dx = (1/√ (1-u ² ))(2x)
dy/dx = (1/√ (1-(x ²) ² ))(2x)
dy/dx = 2x/√ (1-x^4)
At x = 0.5:
dy/dx = 2(0.5)/√ (1-0.5^4) ~ 1.033
Similar approach for 2. and 3, assuming you have the basic inverse trig derivative formulae.
Sure! I'd be happy to explain how to solve each of these differentiation problems step by step.
1. Differentiate y = sin^(-1)(x^2) at x = 0.5:
To differentiate this function, we need to use the chain rule. The chain rule states that if we have a composition of functions, such as f(g(x)), the derivative is given by (d f/dx) = (d f/dg) * (d g/dx).
In this case, g(x) = x^2 and f(x) = sin^(-1)(x). So, let's find the derivative of f(g(x)):
Step 1: Find the derivative of f(x) = sin^(-1)(x):
To find the derivative of arcsin(x), we use the formula d/dx(arcsin(x)) = 1/sqrt(1-x^2). So, (d f/dx) = 1/sqrt(1-x^2).
Step 2: Find the derivative of g(x) = x^2:
To find the derivative of x^2, we use the power rule, which states that (d/dx)(x^n) = n*x^(n-1). In this case, (d g/dx) = 2x.
Step 3: Plug in the values and evaluate at x = 0.5:
Substitute x = 0.5 into both derivatives you found in step 1 and step 2, then multiply them together to get the final derivative: (d f/dx) * (d g/dx) = (1/sqrt(1-(0.5)^2)) * (2*0.5) = (1/sqrt(1-0.25)) * 1 = (1/sqrt(0.75)) ≈ 1.155.
Therefore, the derivative of y = sin^(-1)(x^2) at x = 0.5 is approximately 1.155 (rounded to 3 decimal points).
3. Differentiate y = (x^2 + 1) * tan^(-1)(x) at x = 1:
Again, we will use the chain rule for this problem.
Step 1: Find the derivative of f(x) = tan^(-1)(x):
The derivative of arctan(x) = 1/(1+x^2). So, (d f/dx) = 1/(1+x^2).
Step 2: Find the derivative of g(x) = x^2 + 1:
The derivative of x^2 + 1 is 2x.
Step 3: Plug in the values and evaluate at x = 1:
Substitute x = 1 into both derivatives you found in step 1 and step 2, then multiply them together to get the final derivative: (d f/dx) * (d g/dx) = (1/(1+1^2)) * (2*1) = (1/2) * 2 = 1.
Therefore, the derivative of y = (x^2 + 1) * tan^(-1)(x) at x = 1 is 1.
7. Differentiate y = [cos^(-1)(x)]^3 at x = 0.5:
For this problem, we will once again use the chain rule.
Step 1: Find the derivative of f(x) = [cos^(-1)(x)]^3:
To find the derivative of [cos^(-1)(x)]^3, we need to use the chain rule and the power rule.
The derivative of [cos^(-1)(x)] is -1/(sqrt(1-x^2)), and the derivative of (x)^3 is 3x^2.
Step 2: Plug in the values and evaluate at x = 0.5:
Substitute x = 0.5 into the derivative of [cos^(-1)(x)] and the derivative of (x)^3, then multiply them together to get the final derivative: (d f/dx) * (d g/dx) = [-1/(sqrt(1-(0.5)^2))] * (3*(0.5)^2) = [-1/(sqrt(1-0.25))] * (3*0.25) = [-1/(sqrt(0.75))] * 0.75 = -1 * 0.75 = -0.75.
Therefore, the derivative of y = [cos^(-1)(x)]^3 at x = 0.5 is -0.75.