Derivatives of Trigonometric Function1. y = cos⁴ t - sin⁴ t

To find the derivative of the function y = cos⁴ t - sin⁴ t, we will need to use the chain rule and the power rule.

The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. The power rule states that the derivative of x^n is n*x^(n-1).

Step 1: Identify the outer and inner functions. In this case, for cos⁴ t, the outer function is x⁴ and the inner function is cos t. Similarly, for sin⁴ t, the outer function is x⁴ and the inner function is sin t.

Step 2: Apply the chain rule. The derivative of cos⁴ t is 4cos³ t * -sin t (using the chain rule, where the derivative of cos t is -sin t). The derivative of sin⁴ t is 4sin³ t * cos t (using the chain rule, where the derivative of sin t is cos t).

Step 3: Combine the derivatives. So, the derivative of y = cos⁴ t - sin⁴ t is dy/dt = 4cos³ t * -sin t - 4sin³ t * cos t.

This is the derivative of the given function.

To find the derivative of the function y = cos⁴ t - sin⁴ t, we will need to use the chain rule and the power rule.

The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. The power rule states that the derivative of x^n is n*x^(n-1).

Step 1: Identify the outer and inner functions. In this case, for cos⁴ t, the outer function is x⁴ and the inner function is cos t. For sin⁴ t, the outer function is x⁴ and the inner function is sin t.

Step 2: Apply the chain rule. The derivative of cos⁴ t is 4cos³ t * -sin t (since the derivative of cos t is -sin t). The derivative of sin⁴ t is 4sin³ t * cos t (since the derivative of sin t is cos t).

Step 3: Subtract the two derivatives. The derivative of y = cos⁴ t - sin⁴ t is 4cos³ t * -sin t - 4sin³ t * cos t.

So, the derivative of the function y = cos⁴ t - sin⁴ t is 4cos³ t * -sin t - 4sin³ t * cos t.