Question Video: Finding the Second Derivative of a Function | Nagwa Question Video: Finding the Second Derivative of a Function | Nagwa

# Question Video: Finding the Second Derivative of a Function Mathematics • Third Year of Secondary School

## Join Nagwa Classes

Given that π¦ = (β4π₯ + 7)(β7π₯Β² β 4), determine dΒ²π¦/dπ₯Β².

03:33

### Video Transcript

Given that π¦ is equal to negative four π₯ plus seven multiplied by negative seven π₯ squared minus four, determine d two π¦ by dπ₯ squared.

The question tells us that π¦ is equal to the product of two polynomials, and it wants us to determine d two π¦ by dπ₯ squared. And we know this means the second derivative of π¦ with respect to π₯. Weβre going to need to differentiate this expression twice. Since weβre given π¦ as the product of two functions, we might be tempted to use the product rule, and this would work. However, since our factors are just polynomials with two terms each, itβs actually simpler in this case to just multiply out our factors.

Since weβll just get a polynomial with four terms, weβll multiply these together by using the FOIL method. Letβs start by multiplying the first two terms. We get negative four times negative seven π₯ squared is equal to 28π₯ cubed. Next, we need to multiply our outer terms. This gives us negative four π₯ times negative four, which is equal to 16π₯. Next, we want to multiply our inner two terms. This gives us seven multiplied by negative seven π₯ squared, which is equal to negative 49π₯ squared. Finally, we want to multiply the last two terms in our factors. We get seven times negative four, which is equal to negative 28.

So, we found the following expression for π¦. Itβs a polynomial with four terms. Letβs switch our middle two terms around so weβre writing it in decreasing exponents of π₯. This gives us π¦ is equal to 28π₯ cubed minus 49π₯ squared plus 16π₯ minus 28. Remember, the question wants us to find the second derivative of π¦ with respect to π₯. Weβll start by finding the first derivative of π¦ with respect to π₯. Thatβs dπ¦ by dπ₯, which is equal to the derivative of 28π₯ cubed minus 49π₯ squared plus 16π₯ minus 28 with respect to π₯.

But this is just the derivative of a polynomial. We can do this term by term by using the power rule for differentiation, which tells us for constants π and π, the derivative of ππ₯ to the πth power with respect to π₯ is equal to π times π times π₯ to the power of π minus one. We multiply by the exponent of π₯ and reduce this exponent by one. Using this, weβll differentiate our first term. We get three times 28 times π₯ to the power of three minus one. This simplifies to give us 84π₯ squared. We can also use this to differentiate our second term. We get negative 49 times two, which is negative 98. And then, we have π₯ to the power of two minus one, which is π₯.

We could also differentiate our last two terms by using the power rule for differentiation. However, itβs simpler to notice that these two terms make a linear function. So, their slope will be the coefficient of π₯, which is 16. So, we found an expression for dπ¦ by dπ₯. We can use this to find an expression for our second derivative of π¦ with respect to π₯. Our second derivative of π¦ with respect to π₯ would just be the derivative of dπ¦ by dπ₯ with respect to π₯.

So, to find d two π¦ by dπ₯ squared, weβre going to differentiate dπ¦ by dπ₯ with respect to π₯. Thatβs the derivative of 84π₯ squared minus 98π₯ plus 16 with respect to π₯. And again, this is the derivative of a polynomial. So, we can do this by using the power rule for differentiation. Our first term will be two times 84 times π₯ to the first power, which is 168π₯. And our second term will just be the coefficient of π₯, which is 98.

Therefore, weβve shown if π¦ is equal to negative four π₯ plus seven times negative seven π₯ squared minus four, then d two π¦ by dπ₯ squared is equal to 168π₯ minus 98.

## Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions