Question

Calculate the gradient of the curve at the point where \[x = 1\] and \[y = x^{3} + 7\] ?

Answer 1

Hint: In this question, we need to find the gradient of the given curve \[y = x^{3} + 7\] and also given that the gradient is at the point \[x\ = 1\ \] . The gradient of a line is defined as the measure of its Steepness. It is calculated by dividing the change in y coordinate by change in x coordinate. Mathematically, gradient is denoted by the letter \[m\] .Here the gradient of the curve can be found from the differentiation of the given curve.First, we need to differentiate the given expression by applying the derivative power rule and need to be careful in using the power rule. After differentiating the curve , we need to substitute the given point in the differentiated equation. With the help of derivation and derivative rules, we can find the gradient of the curve.
Derivative rules used :
1.\[\dfrac{\text{d}}{\text{dx}}\left( x^{n} \right) = nx^{n – 1}\]
2.\[\dfrac{\text{d}}{\text{dx}}(k) = 0\]

Complete answer:
Given, curve \[y = x^{3} + 7\]
The gradient of the curve can be found by differentiating the given curve.
\[y = x^{3} + 7\]
On differentiating both sides,
We get,
\[\Rightarrow \dfrac{\text{dy}}{\text{dx}} = 3x^{2} + 0\]
Also given that the gradient at \[x = 1\] ,
By substituting the value of \[x\] ,
We get,
\[\Rightarrow \dfrac{\text{dy}}{\text{dx}} = 3\left( 1 \right)^{2}\]
On simplifying,
We get,
\[\dfrac{\text{dy}}{\text{dx}} = 3\]
Thus we get the gradient of the curve at the point where \[x = 1\] and \[y = x^{3} + 7\] is \[3\]
The gradient of the curve at the point where \[x = 1\] and \[y = x^{3} + 7\] is \[3\].

Note:
The gradient is positive when \[m\] is greater than \[0\] and when \[m\] is less than \[0\] , then the gradient is negative. If the gradient is equal to \[0\] That means it is a constant function. Graphically, The gradient of two parallel lines is equal and also the product of the gradients of two perpendicular lines is \[-1\]. Mathematically, a derivative is defined as a rate of change of function with respect to an independent variable given in the function. The term differentiation is nothing but it is a process of determining the derivative of a function at any point. Also, while differentiating we should be careful in using the power rule \[\dfrac{\text{d}}{\text{dx}}\left( x^{n} \right) = nx^{n – 1}\] , a simple error that may happen while calculating.