Find d𝑦 by d𝑥 if 𝑦 equals two 𝑥 to the power of negative seven.
In this question, we were asked to find d𝑦 by d𝑥, which we recall means the first derivative of 𝑦 with respect to 𝑥. So we’re going to answer this question using differentiation. When we look at our function 𝑦, we see that it consists of just a single term in 𝑥. It is a general polynomial term, two multiplied by 𝑥 to the power of negative seven.
We therefore need to recall the power rule of differentiation, which tells us how to differentiate general polynomial terms. In its most straightforward form, the power rule tells us that the derivative with respect 𝑥 of 𝑥 to some power 𝑛, where 𝑛 is a real number, is equal to 𝑛 multiplied by 𝑥 to the power of 𝑛 minus one. We multiply by the original exponent and then reduce the exponent by one.
If we introduce the coefficient 𝑎, so we’re looking to find the derivative with respect to 𝑥 of 𝑎𝑥 to the 𝑛th power. Then we recall that the rules of differentiation tell us that this is just equal to 𝑎 multiplied by the derivative with respect 𝑥 of 𝑥 to the 𝑛th power. So by applying the first rule, this is equal to 𝑎𝑛𝑥 to the power of 𝑛 minus one.
In this question then, the value of 𝑎, the coefficient, is two and the value of 𝑛, the exponent, is negative seven. Let’s apply the power rule of differentiation then. We have that d𝑦 by d𝑥 will be equal to two. Then we multiply by the original exponent, which was negative seven. And then we reduce the exponent by one, giving 𝑥 to the power of negative seven minus one.
Simplifying the coefficient, two multiplied by negative seven is negative 14. And then simplifying the exponent, negative seven minus one is negative eight. So we have that d𝑦 by d𝑥 is equal to negative 14 multiplied by 𝑥 to the power of negative eight.
Be careful here because we’re working with a negative exponent. And remember, we need to reduce the exponent by one. A common mistake would be to make the new exponent negative six rather than negative eight.
So by applying the power rule of differentiation, we’ve found that if 𝑦 is equal to two 𝑥 to the power of negative seven, then d𝑦 by d𝑥 is equal to negative 14𝑥 to the power of negative eight.