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

Question

𝔂 = 3 .

(5x² + sin 2x)³/²

Accepted Answer

Welcome back, everyone. Calculate the derivative of the function F of X equals 5 divided by 2 X cubed minus 3X, raise the power of 1/3. What we're going to do is identify the derivative of F of X, specifically, we're evaluating F of X. First of all, let's rewrite the function S5 multiplied by the denominator, which is 2 X cubed minus sine of 3 X, raises the power of -13. So here we are applying the properties of exponents. If we have a denominator, we can rewrite it as a product instead of a division, right? If we take the negative of the original exponent instead of 1/3, we're taking negative 1/3, and this allows us to apply the power rule. We can factor out the constant. That's 5, and now this means that we're differentiating our function 2 X cubed minus sine of 3 X raises the power of -1/3. We understand that our function is a composite function. So first of all, what we're going to do is apply the power rule. So we are bringing our exponent down, meaning we're multiplying by 1/3. The original exponent of 2 X cubed minus sine of 3 X decreases by 1 unit, so -13 minus 1 gives us -43. And then according to the chain rule, we have to multiply the result by the derivative of the inner function, which is 2 X cub minus sine of 3 X, right? So now what we're going to do is simply evaluate the final derivative. Let's differentiate 2 X cubed minus sine of 3 X. We have two terms. The derivative of the first one can be identified using the power rule. The derivative of X cubed is 3x2d and 3 X squared multiplied by 2 is 6X2. Now the derivative of sine of 3 x is cosine of 3x based on the table of basic derivatives, and because once again we have a composite function based on the chain rule we're multiplying by the derivative of the inner function, which is 3 X. Let's simplify. We're going to get 6 X squared. Minus now the derivative of 3 X is simply 3, so we are subtracting 3 cosine of 3 X. So now let's go ahead and use these results in the original expression of F of X. We have 5 multiplied by -13, which is 5/3. Multiplied by 2 X cubed minus sine of 3 X raise the power of -4/3 multiplied by. Now we can factor out 6 X2 minus 3 cosine of 3 X, we can take out 3, and this leaves us with 2 X2 minus cosine of 3 X. This is really useful because we have a denominator of 3, and this allows us to cancel out those 3s. So now let's write the final answer. First of all, we have a fraction based on the fact that we have a number or specifically a function raised to a negative exponent, right? So we can turn it into a positive exponent if we write it in the denominator. So this gives us 2 X cubed minus sign of 3 x in the denominator race 2. 4/3, right? Now we are turning it into a positive exponent, and now in the numerator, we have 5. We have already added the negative sign and in 2 X2 minus cosine of 3 X. And that's it. We have identified the final derivative for this problem. Thank you for watching.

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

𝔂 = 3 .
(5x² + sin 2x)³/²

Key Concepts

Derivative

Chain Rule

Power Rule

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