Question

Find the slope of the tangent to the curve $y = 3{x^4} - 4x$ at $x = 4$

Answer 1

Hint: This sum is from the chapter curves. While solving sums like these , one should know chapters on derivatives. When it is asked to find the slope, we will compulsorily have to use the first derivative of the equation of the curve. After finding the derivative we need to put a value of $x$ at which we have to find the slope.

Complete step-by-step answer:
We have to find the slope of the tangent to the curve$y = 3{x^4} - 4x$.
Tangent to this curve at $x = 4$ would satisfy the equation of the curve . Thus we will have to find the slope of the curve. Since it is the tangent, the slope of the curve and the slope of the tangent would be the same.
We find the derivative of $y = 3{x^4} - 4x$ with respect to $x$
$\therefore \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}(3{x^4} - 4x)$
Solving the derivative we get
$\therefore \dfrac{{dy}}{{dx}} = 12{x^3} - 4..........(1)$
It is given that we have to find the slope of tangent at $x = 4$.
We substitute the value of $x = 4$ in equation $1$
$\therefore \dfrac{{dy}}{{dx}} = 12{(4)^3} - 4$
$\therefore \dfrac{{dy}}{{dx}} = 764$

$\therefore $Slope of the tangent to the curve is $764$.

Note: Since this is an easy sum, students should not mess up while calculating the derivative. It is necessary to know the chapter on derivative while solving problems for this chapter or any chapter involving computation of slope. All the formulae for derivatives should be learnt by heart. Since it is a one-step sum, students are advised to show each and every calculation done. Chapter on curves involve application of derivatives and integration, so that students should always remember the basic formulae. Also the student should immediately recognize the type of curve based on the equation given and then proceed with the sum.