If f(t)=t¹⁰, find f′(t), f′′(t), and f′′′(t).

Question

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says, given the function G of X is equal to X-rays to the 7th power, determine G of X, G of X, and G of X. So, this question is asking us to calculate various derivatives for our function GFX. And so the first quantity that we're going to calculate is G X. And this is going to be equal to the derivative with respect to X. Of our function G of X. So here we're calculating the first derivative. And so this is going to be equal to the derivative with respect to X. Of the quantity of X-rays to the 7th power. Now, to take the derivative of this expression, we're going to need to use our power rule for derivatives. So recall for that this is going to be equal to our Xponent, which is 7, multiplied by x raised to the quantity of 7 minus 1. And simplifying this expression, this is going to be equal to 7 multiplied by x raised to the 6th power. And so this is our answer for our first derivative G of X. Now, for the next part of the problem, we need to calculate the second derivative of G with respect to X, so we're looking for G of X. And this is going to be equal to the derivative with respect to X of our first derivative function G of X. So this is going to be equal to the derivative with respect to X of the quantity of the expression that we found for G of X, which is 7 multiplied by X rated to the 6th power. Again, we're going to apply our power rule to calculate this derivative. This is going to be equal to 7, multiplied by the exponent, which is 6, multiplied by X, raised to the quantity of 6 minus 1. So simplifying this expression, this is going to be equal to 42, multiplied by x raised to the 5th power. And so this is the answer for our second derivative of G with respect to X. So for G of Xs. Now, the final quantity that we need to calculate is G of X, and this is going to be equal to the derivative with respect to X of our our function G of X, so it's gonna be the derivative of our second derivative function. So this is going to be equal to the derivative with respect to X of the quantity of 42, multiplied by X-rays to the 5th power. And again, we'll apply our power rule to take this derivative. This is going to be equal to 42, multiplied by 5, multiplied by X raised to the quantity of 5 minus 1. So again, simplifying this expression, this is going to be equal to 210 multiplied by X raised to the 4th power. And so that is our answer for G of Xs. So here we've calculated all of the derivatives that we were asked um to in this problem, so this is the final answer for our problem. So, I hope that this explanation has been useful, and I'll see you in the next video.

If f(t)=t¹⁰, find f′(t), f′′(t), and f′′′(t).

Key Concepts

Differentiation

Power Rule

Higher-Order Derivatives

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