Find the derivatives of the functions in Exercises 1–42.

Question

s = cos⁴ (1 - 2t)

Accepted Answer

Welcome back everyone. Calculate the derivative of the function F of X equals sine rate to the 5th of. We X + 2. For this problem we want to valid F of X, which is simply the derivative of the original function. This means that we're differentiating sin to the fifth of V X + 2. We have to recognize that we have a composite function because there's an exponent adjacent to sign, so that's our outer function, and the inner function is sign of 3 x + 2. This means that we're applying the chain rule. According to the chain rule, we are first of all taking care of the outer function and because it is an exponent, we have to apply the power rule. The power rule says that we're going to bring down the 5, so that 5 becomes our coefficient. We're multiplying 5 by the inner function, which is sign of P X + 2. And now the original exponent decreases by 1 unit, so 5 minus 1 gives us 4. So now sine has an exponent of 4. According to the chain rule, we have to multiply the result by the derivative of the inner function, and because our inner function is sin of 3 x + 2, we're simply taking the derivative of this term. So let's continue. We're going to rewrite the first term by assigned to the 4th. Of 3 X + 2. And now we have to notice that our derivative is also a composite function. It has sine in the inner function 3X + 2. So first of all, we're differentiating it with respect to the inner function 3X + 2, and we know that the derivative of sine is cosine based on the table of basic derivatives. So we get cosine of the original inner function 3 x + 2. In applying the chain rule once again, we're multiplying by the derivative of the inner function, so the derivative of 3 x + 2. And now we can evaluate the final results. We get 5 sign to the 4th. Of 3 x + 2 multiplied by cosine of 3 x + 2 multiplied by the derivative of 33 based on the fact that the derivative of X is 1 and the derivative of 2 is 0 because it's a constant. So now we can combine 5 and 3 because 5 multiplied by 3 is 15, and our final answer is 15 signed to the 4th of 3 X + 2. Multiplied by cosine of 3 x + 2. That would be our final answer, and thank you for watching.

Find the derivatives of the functions in Exercises 1–42.

s = cos⁴ (1 - 2t)

Key Concepts

Chain Rule

Power Rule

Trigonometric Derivatives

Watch next