Video: Finding the Slope of the Tangent to the Curve of a Polynomial Function at a Given 𝑥-Coordinate

Find the slope of the tangent to the curve 𝑦 = 4𝑥³ + 4𝑥 at 𝑥 = 1.

02:20

Video Transcript

Find the slope of the tangent to the curve 𝑦 equals four 𝑥 cubed plus four 𝑥 at 𝑥 equals one.

So if we’ve got the curve 𝑦 equals four 𝑥 cubed plus four 𝑥, then the slope of the tangent to this curve at the point 𝑥 equals one is going to be equal to the slope of the curve at that point. So therefore, to enable us to find the slope of our curve, what we need to do is find the slope function of our curve. And we call that d𝑦 d𝑥. And to find this, what we’re going to do is differentiate four 𝑥 cubed plus four 𝑥.

So therefore, the first term of our slope function is gonna be 12𝑥 squared. And we get that because we multiplied the exponent by the coefficients, so three multiplied by four, which gives us 12. And then, we reduce the exponent by one. So we get 12𝑥 squared. And then, the second term is just gonna be positive four. And that’s because if we multiply the exponent of 𝑥, well that would be one. So one multiplied by four gives us four. And then we have 𝑥 to the power of — well, one minus one is zero. Anything to the power of zero is just one. So we’re left with four.

So great, we now have our slope function. But what we want to do is find the slope of the tangent to the curve 𝑦 equals four 𝑥 cubed plus four 𝑥 at 𝑥 equals one. Well, as we’ve already stated, the slope to the tangent to the curve at this point is going to be the same as the slope of the curve at this point. So what we’re going to do is substitute 𝑥 equals one into our slope function. And when we do that, we’re gonna get the slope is equal to 12 multiplied by one squared plus four is gonna be equal to 16. That’s because we have 12 add four, which is 16.

So therefore, we can say that the slope to the tangent to the curve 𝑦 equals four 𝑥 cubed plus four 𝑥 at 𝑥 equals one is going to also be 16. And that’s because, remembering what we said earlier, the slope of a tangent at a point is equal to the slope of the curve at that same point.

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