Find the function f given that the slope of the tangent line to the graph of f at any point f at

Find the function f given that the slope of the tangent line...

Key Concepts

Derivative

The derivative is a fundamental concept in calculus that represents the instantaneous rate of change of a function at a specific point. It describes how the function's output changes with respect to changes in the input, and in this problem, it is given by the expression x?(x+1), which corresponds to the slope of the tangent line at any point on the function's graph.

Indefinite Integration

Indefinite integration, or finding the antiderivative, is the reverse process of differentiation used to recover a function from its derivative. By integrating the given derivative x?(x+1), we obtain a family of functions that differ only by a constant, which represents the set of all potential antiderivatives.

Initial Condition

An initial condition is a specific value of the function at a given point, used to determine the unique solution from the general family of solutions obtained through integration. In this case, the point (3,6) is provided so that the constant of integration can be found, ensuring that the function not only satisfies the derivative condition but also passes through the specified point.

Transcript

00:01 In the question given data is there is a slope of tangent line means m is equal to given as x times of square root of x plus 1 and given point is that is whose coordinates are 3 .6 so by these two data i have to find required function f equation so for finding this first i will write the given slope that is m is equal to x times of square root of x plus 1 and we know that slope will be equal to d .y by d x so i can say that d .y by d x will be equal to x times of square root of x plus 1 now again it will be written as that dy will be equal to x times of square root of x plus 1 d x now take integration on both sides so on the left side after integration it will become y and on the right side after integration it will be written as integration so integral x times of square root of x plus 1 d x now my next step is trying to find this integral so first, i will again assume that u will be equal to let u is equal to x plus 1.

01:47 And so, so x will give value that is equal to u minus 1.

01:56 So the y will be written in term of u that is y is equal to integral x means that is u minus 1 times of square root of x plus 1 means u and the d x will be equal to as you see x is equal to u minus 1 taking differentiation on both side the value of d x will be equal to d u so i can say that put here value of d x that is d u so now y will be equal to integral u minus 1 square root of u d u d u now my next step is that multiply root u inside the bracket so i will get that y will be equal to after multiplying i will get that is u to the power 3 over 2 minus u to the power half d u now i am integrating every term so i will get the value of why that is after indication of u to the power 3 over 2 d u value that is 2 divided by 5 times of u to the power 5 over 2 minus indication of u to the power half d u will be 2 divided by 3 times of u to the power 3 over 2 plus c is a constant value now this is the function equation but here we have to find the value of c.

03:46 So for finding c, i will use the second data which is saying that this graph function will pass through given point that is 3 .6.

04:00 So put the value of x and y that is 3 and 6 in the above equation.

04:07 So i will get the value of c...

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