The value of the given differentiation f'(1).f'(1)=1024/1089 if f(x)+x²[f(x)]⁴=18.
<h3>What is meant by differentiation?</h3>
To ascertain the instantaneous rate of change of a function dependent on one of its variables, differentiation is applied. The most common example is velocity, which is the rate at which a distance changes with respect to time.
The derivative of a function of a real variable in mathematics is used to assess a function's sensitivity to change with respect to a change in its argument. The derivative is a key mathematics tool.
The slope of the tangent line to the graph of a single-variable function at a specific input value is referred to as the derivative if the function exists at that point. The tangent line is the most accurate linear approximation of the function close to that input value.
Given,
f(x)+x²[f(x)]⁴=18
Now, by differentiating on both sides, we get:
f'(x)+2x[f(x)]⁴+4x²[f(x)]³f'(x)=0
f'(x)[1+4x²[f(x)]³]=-2x[f(x)]⁴
f'(x)= (-2x[f(x)]⁴)/[1+4x²[f(x)]³]
Given,
f(1)=2
Let x=1 then,
f'(1)=(-2(1)[f(1)]⁴)/[1+4x²[f(1)]³]
f'(1)=(-2×2⁴)/(1+4×2³)
f'(1)=-32/33
f'(1).f'(1)=(-32/33)(-32/33)
=1024/1089
Therefore, the value of the given differentiation f'(1).f'(1)=1024/1089 if f(x)+x²[f(x)]⁴=18.
To know more about differentiation, visit:
brainly.com/question/24898810
#SPJ1
