find derivative of cos^4(x)^4
To find the derivative of cos^4(x)^4, we can use the chain rule and the power rule. Here is how you can do it step by step:
Step 1: Rewrite the expression using the power rule for exponents. Recall that raising a power to another power, you need to multiply the exponents. We have: (cos^4(x))^4 = cos^(4*4)(x) = cos^16(x).
Step 2: Apply the chain rule. The chain rule states that if we have a composite function, f(g(x)), then its derivative is given by f'(g(x)) * g'(x). In this case, f(u) = u^16, and g(x) = cos(x).
Step 3: Find the derivative of f(u) = u^16. We can use the power rule, which states that the derivative of u^n, where n is a constant, is given by n*u^(n-1). Applying this rule, the derivative of f(u) = u^16 is f'(u) = 16u^15.
Step 4: Find the derivative of g(x) = cos(x). The derivative of cos(x) is -sin(x), so g'(x) = -sin(x).
Step 5: Apply the chain rule to f(u) = u^16 and g(x) = cos(x). By multiplying f'(g(x)) and g'(x) together, we obtain (-sin(x)) * (16cos^15(x)) as the derivative of cos^16(x).
Therefore, the derivative of cos^4(x)^4, or cos^16(x), is (-sin(x)) * (16cos^15(x)).