Given 𝑦 equals sin four 𝑥 tan four 𝑥, find d𝑦 by d𝑥 at 𝑥 equals 𝜋 by six.
In this question, we’re asked to find d𝑦 by d𝑥, that’s the first derivative of this function 𝑦, and then evaluate it at a particular 𝑥-value, the 𝑥-value 𝜋 by six. We can see that the function we’ve been given is the product of two functions. It’s equal to sin four 𝑥 multiplied by tan four 𝑥. And so, in order to find its derivative, we’re going to need to apply the product rule.
This tells us that for two differentiable functions 𝑢 and 𝑣, the derivative with respect to 𝑥 of their product 𝑢𝑣 is equal to 𝑢 times d𝑣 by d𝑥 plus 𝑣 times d𝑢 by d𝑥. We multiply each function by the derivative of the other and add them together. We can, therefore, define 𝑢 to be the function sin four 𝑥 and 𝑣 to be the second factor, 𝑣 is equal to tan four 𝑥. We then need to find each of their individual derivatives. And to do so, we’re going to need to recall some standard results for differentiation of trigonometric functions.
Firstly, we recall that the derivative with respect to 𝑥 of sin of 𝑎𝑥 for some constant 𝑎 is equal to 𝑎 multiplied by cos of 𝑎𝑥. And this can be proven from first principles. Although we must remember that these results for trigonometric functions are only valid if the angle is measured in radians. Secondly, we recall that the derivative with respect to 𝑥 of tan 𝑎𝑥 is equal to 𝑎 multiplied by sec squared 𝑎𝑥. And we can prove this by recalling that tan of 𝑎𝑥 is equal to sin 𝑎𝑥 over cos 𝑎𝑥. We can then recall the standard results for differentiating sin of 𝑎𝑥 and cos of 𝑎𝑥 and apply the quotient rule.
In each case here, the value of the constant 𝑎 is four. So, we have that the derivative of sin four 𝑥 with respect to 𝑥 is four cos four 𝑥 and that the derivative of tan four 𝑥 with respect to 𝑥 is four sec squared four 𝑥. Now, we’re ready to substitute into the product. Well, we have that d𝑦 by d𝑥 is equal to 𝑢 times d𝑣 by d𝑥, that’s sin four 𝑥 multiplied by four sec squared four 𝑥, plus 𝑣 times d𝑢 by d𝑥, that’s tan four 𝑥 multiplied by four cos four 𝑥.
We can then manipulate this expression slightly. In the first term, we have sec squared four 𝑥, which is equal to one over cos squared four 𝑥. So, we can write the first term as four sin four 𝑥 over cos four 𝑥 multiplied by one over cos four 𝑥. In the second term, we have tan four 𝑥, which can be written as sin of four 𝑥 over cos four 𝑥. So, the second term becomes four sin four 𝑥 over cos four 𝑥 multiplied by cos four 𝑥.
Within the second term then, we see that a factor of cos four 𝑥 can be cancelled from both the numerator and denominator to leave the simplified term four sin four 𝑥. And in the first term, where we have sin four 𝑥 over cos four 𝑥, we could rewrite this as tan four 𝑥. So, our simplified expression for d𝑦 by d𝑥 is four tan four 𝑥 over cos four 𝑥 plus four sin four 𝑥.
Next, we need to evaluate this expression when 𝑥 is equal to 𝜋 by six. Four 𝑥 will, therefore, be equal to four 𝜋 by six, which simplifies to two 𝜋 by three. When 𝑥 is equal to 𝜋 by six then, d𝑦 by d𝑥 will be equal to four tan two 𝜋 by three over cos of two 𝜋 by three plus four sin two 𝜋 by three. Now, if we have a calculator, we can evaluate this directly. But if not, we can still answer the question. Because two 𝜋 by three is one of those special angles for which we need to know the values of its trigonometric ratios of by heart.
We should recall that tan of two 𝜋 by three is equal to negative root three and cos of two 𝜋 by three is equal to negative one-half. In the same way, sin of two 𝜋 by three is equal to root three over two. So, we have exact values for each of these trigonometric ratios. We, therefore, have negative four root three over negative a half, which is the same as negative four root three multiplied by negative two, plus four root three over two.
The first term here will simplify to eight root three. And in the second term, we can cancel a common factor of two to leave us with two root three. So, we have that d𝑦 by d𝑥, when 𝑥 equals 𝜋 by six, is equal to eight root three plus two root three, which is of course 10 root three. So, we’ve applied the product rule and standard results for differentiation of trigonometric functions to find that if 𝑦 is the function sin four 𝑥 multiplied by tan four 𝑥, then its first derivative d𝑦 by d𝑥 at 𝑥 equals 𝜋 by six is 10 root three.